introduction - matrixology (linear algebra)—episode 1/24 math … · 2016-08-29 · episode1/24:...
TRANSCRIPT
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
1 of 45
IntroductionMatrixology (Linear Algebra)mdashEpisode 124
MATH 122 Fall 2016
Prof Peter Dodds
Dept of Mathematics amp Statistics | Vermont Complex Systems CenterVermont Advanced Computing Core | University of Vermont
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 30 License
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
2 of 45
These slides are brought to you by
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
3 of 45
Outline
Exciting Admin
Importance
Usages
Key problems
Three ways of looking
Colbert on Equations
References
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
4 of 45
Basics
Instructor Prof Peter Dodds Lecture room and meeting timesPerkins 107Tuesday and Thursday 1005 am to 1120 am Office Farrell Hall second floor Trinity Campus E-mail peterdoddsuvmedu Course websitehttpwwwuvmedu pdoddsteachingcourses2016-08UVM-122 Textbook ldquoIntroduction to Linear Algebrardquo (3rd or4th or 5th edition) by Gilbert Strang (published byWellesley-Cambridge Press)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
5 of 45
Our Textbook of Excellence
4th Edition V 3rd Edition V Unhelpful 2 ldquoIntroduction to Linear Algebrardquoby Gil Strang Textbook websitehttpmathmitedulinearalgebra MIT Open Courseware site for 1806(=Linear Algebra)httpocwmitedulinear-algebra-spring-2010
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
6 of 45
Yesness
Money quote from George Cobbrsquos review ofStrangrsquos bookDo you want a book written by a mathematician with alifetime experience using linear algebra to understandimportant authentic applied problems a formerpresident of the Society for Industrial and AppliedMathematics hellip
or do you want a book shaped mainly by the[a]esthetics of pure mathematicians with only a weaktheoretical connection to how linear algebra is used inthe natural and social sciences George Cobb Robert L Rooke Professor of
Mathematics and Statistics Mount Holyoke College Full review here [amazon]
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
7 of 45
Gil Strang Exalted Friend of the Matrix Professor of Mathematics at MIT since 1962 Many awardsincluding MAAHaimo Award forDistinguishedCollege orUniversity Teachingof Mathematics Rhodes Scholar Legend
More on Laplacian matrices graphs and othermadnesses here (Strangrsquos Wikipedia page is here
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
8 of 45
Admin
Potential paper products1 Outline
Papers to read1 ldquoThe Fundamental Theorem of Linear Algebrardquo [2]
2 ldquoToo Much Calculusrdquo [3]
Office hours 1000 to 1155 am WednesdaysFarrell Hall second floor Trinity Campus
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
9 of 45
Team MatrixologyWe may try out Slack Place for discussions about all things PoCS
including assignments and projects Once invited please sign up herehttpteam-matrixologyslackcom Very good Install Slack app on laptops tabletsphone Everyone will behave wonderfully
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
10 of 45
Grading breakdown
1 Levels (40) Ten one-week assignments Lowest assignment score will be dropped The last assignment cannot be dropped Each assignment will have a random bonus pointquestion which has nothing to do with linearalgebra
2 Challenge Levels (30) Three 75 minutes tests distributed throughout thecourse all of equal weighting
3 Final Boss Level (20) le Three hours of joyful celebration Thursday December 15 130 pm to 415 pm inPerkins 107
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
11 of 45
Grading breakdown
4 Mini-levels (10) Most meeting times will end with a 10 to 15minute mini-level There will be around 20 mini-levels
5 Homework (0)mdashProblems assigned online fromthe textbook Doing these exercises will be mostbeneficial and will increase happiness
6 General existencemdashit is extremely desirable thatstudents attend class and class presence will betaken into account if a grade is borderline
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
12 of 45
Questions are worth 3 points according to thefollowing scale 3 = correct or very nearly so 2 = acceptable but needs some revisions 1 = needs major revisions 0 = way off
Schedule The course will mainly cover chapters 2 through 6of the textbook (You should know all about Chapter 1)
Week (dates) Tuesday Thursday1 (830 and 901) = = + Level 12 (906 and 908) = = + Level 23 (913 and 915) = = + Level 34 (920 and 922) = and review Challenge Level 15 (927 and 929) Big picture Big picture + Level 46 (1004 and 1006) Big picture Big picture + Level 57 (1011 and 1013) Big picture Big picture + Level 68 (1018 and 1020) Big picture Challenge Level 29 (1025 and 1027) Normal equation Gram-Schmidt Process +
Level 710 (1101 and 1103) Eigenstuff Eigenstuff + Level 811 (1108 and 1110) Determinants Determinants + Level 912 (1115 and 1117) Eigenstuff textitChallenge Level 313 (1122 and 1124) Thanksgiving Thanksgiving14 (1129 and 1201) Positive Definite Matrices
+ Level 10SVD
15 (1206) SVD mdash
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
14 of 45
Important dates1 Classes run from Tuesday August 30 to Tuesday
December 62 AddDrop Audit PassNo Pass
deadlinemdashMonday September 123 Last day to withdrawmdashMonday October 31
(Sadness)4 Reading and Exam periodmdashSaturday December
10 to Friday December 16
More stuffDo check your zoo account for updates regarding thecourse
Academic assistance Anyone who requires assistancein any way (as per the ACCESS program or due toathletic endeavors) please see or contact me as soonas possible
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
15 of 45
More stuff
Being good people1 In class there will be no electronic gadgetry no cell
phones no beeping no text messaging etc Youreally just need your brain some paper and awriting implement here (okay and Matlab)
2 Second I encourage you to email me questionsideas comments etc about the class but requestthat you please do so in a respectful fashion
3 Finally as in all UVM classes Academic honestywill be expected and departures will be dealt withappropriately See httpwwwuvmeducses forguidelines
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
16 of 45
Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0
Computing Approximately 2 out of 10 questions perassignment will be Matlab based
Note for assignment problems written details ofcalculations will be required
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
2 of 45
These slides are brought to you by
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
3 of 45
Outline
Exciting Admin
Importance
Usages
Key problems
Three ways of looking
Colbert on Equations
References
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
4 of 45
Basics
Instructor Prof Peter Dodds Lecture room and meeting timesPerkins 107Tuesday and Thursday 1005 am to 1120 am Office Farrell Hall second floor Trinity Campus E-mail peterdoddsuvmedu Course websitehttpwwwuvmedu pdoddsteachingcourses2016-08UVM-122 Textbook ldquoIntroduction to Linear Algebrardquo (3rd or4th or 5th edition) by Gilbert Strang (published byWellesley-Cambridge Press)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
5 of 45
Our Textbook of Excellence
4th Edition V 3rd Edition V Unhelpful 2 ldquoIntroduction to Linear Algebrardquoby Gil Strang Textbook websitehttpmathmitedulinearalgebra MIT Open Courseware site for 1806(=Linear Algebra)httpocwmitedulinear-algebra-spring-2010
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
6 of 45
Yesness
Money quote from George Cobbrsquos review ofStrangrsquos bookDo you want a book written by a mathematician with alifetime experience using linear algebra to understandimportant authentic applied problems a formerpresident of the Society for Industrial and AppliedMathematics hellip
or do you want a book shaped mainly by the[a]esthetics of pure mathematicians with only a weaktheoretical connection to how linear algebra is used inthe natural and social sciences George Cobb Robert L Rooke Professor of
Mathematics and Statistics Mount Holyoke College Full review here [amazon]
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
7 of 45
Gil Strang Exalted Friend of the Matrix Professor of Mathematics at MIT since 1962 Many awardsincluding MAAHaimo Award forDistinguishedCollege orUniversity Teachingof Mathematics Rhodes Scholar Legend
More on Laplacian matrices graphs and othermadnesses here (Strangrsquos Wikipedia page is here
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
8 of 45
Admin
Potential paper products1 Outline
Papers to read1 ldquoThe Fundamental Theorem of Linear Algebrardquo [2]
2 ldquoToo Much Calculusrdquo [3]
Office hours 1000 to 1155 am WednesdaysFarrell Hall second floor Trinity Campus
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
9 of 45
Team MatrixologyWe may try out Slack Place for discussions about all things PoCS
including assignments and projects Once invited please sign up herehttpteam-matrixologyslackcom Very good Install Slack app on laptops tabletsphone Everyone will behave wonderfully
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
10 of 45
Grading breakdown
1 Levels (40) Ten one-week assignments Lowest assignment score will be dropped The last assignment cannot be dropped Each assignment will have a random bonus pointquestion which has nothing to do with linearalgebra
2 Challenge Levels (30) Three 75 minutes tests distributed throughout thecourse all of equal weighting
3 Final Boss Level (20) le Three hours of joyful celebration Thursday December 15 130 pm to 415 pm inPerkins 107
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
11 of 45
Grading breakdown
4 Mini-levels (10) Most meeting times will end with a 10 to 15minute mini-level There will be around 20 mini-levels
5 Homework (0)mdashProblems assigned online fromthe textbook Doing these exercises will be mostbeneficial and will increase happiness
6 General existencemdashit is extremely desirable thatstudents attend class and class presence will betaken into account if a grade is borderline
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
12 of 45
Questions are worth 3 points according to thefollowing scale 3 = correct or very nearly so 2 = acceptable but needs some revisions 1 = needs major revisions 0 = way off
Schedule The course will mainly cover chapters 2 through 6of the textbook (You should know all about Chapter 1)
Week (dates) Tuesday Thursday1 (830 and 901) = = + Level 12 (906 and 908) = = + Level 23 (913 and 915) = = + Level 34 (920 and 922) = and review Challenge Level 15 (927 and 929) Big picture Big picture + Level 46 (1004 and 1006) Big picture Big picture + Level 57 (1011 and 1013) Big picture Big picture + Level 68 (1018 and 1020) Big picture Challenge Level 29 (1025 and 1027) Normal equation Gram-Schmidt Process +
Level 710 (1101 and 1103) Eigenstuff Eigenstuff + Level 811 (1108 and 1110) Determinants Determinants + Level 912 (1115 and 1117) Eigenstuff textitChallenge Level 313 (1122 and 1124) Thanksgiving Thanksgiving14 (1129 and 1201) Positive Definite Matrices
+ Level 10SVD
15 (1206) SVD mdash
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
14 of 45
Important dates1 Classes run from Tuesday August 30 to Tuesday
December 62 AddDrop Audit PassNo Pass
deadlinemdashMonday September 123 Last day to withdrawmdashMonday October 31
(Sadness)4 Reading and Exam periodmdashSaturday December
10 to Friday December 16
More stuffDo check your zoo account for updates regarding thecourse
Academic assistance Anyone who requires assistancein any way (as per the ACCESS program or due toathletic endeavors) please see or contact me as soonas possible
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
15 of 45
More stuff
Being good people1 In class there will be no electronic gadgetry no cell
phones no beeping no text messaging etc Youreally just need your brain some paper and awriting implement here (okay and Matlab)
2 Second I encourage you to email me questionsideas comments etc about the class but requestthat you please do so in a respectful fashion
3 Finally as in all UVM classes Academic honestywill be expected and departures will be dealt withappropriately See httpwwwuvmeducses forguidelines
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
16 of 45
Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0
Computing Approximately 2 out of 10 questions perassignment will be Matlab based
Note for assignment problems written details ofcalculations will be required
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
3 of 45
Outline
Exciting Admin
Importance
Usages
Key problems
Three ways of looking
Colbert on Equations
References
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
4 of 45
Basics
Instructor Prof Peter Dodds Lecture room and meeting timesPerkins 107Tuesday and Thursday 1005 am to 1120 am Office Farrell Hall second floor Trinity Campus E-mail peterdoddsuvmedu Course websitehttpwwwuvmedu pdoddsteachingcourses2016-08UVM-122 Textbook ldquoIntroduction to Linear Algebrardquo (3rd or4th or 5th edition) by Gilbert Strang (published byWellesley-Cambridge Press)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
5 of 45
Our Textbook of Excellence
4th Edition V 3rd Edition V Unhelpful 2 ldquoIntroduction to Linear Algebrardquoby Gil Strang Textbook websitehttpmathmitedulinearalgebra MIT Open Courseware site for 1806(=Linear Algebra)httpocwmitedulinear-algebra-spring-2010
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
6 of 45
Yesness
Money quote from George Cobbrsquos review ofStrangrsquos bookDo you want a book written by a mathematician with alifetime experience using linear algebra to understandimportant authentic applied problems a formerpresident of the Society for Industrial and AppliedMathematics hellip
or do you want a book shaped mainly by the[a]esthetics of pure mathematicians with only a weaktheoretical connection to how linear algebra is used inthe natural and social sciences George Cobb Robert L Rooke Professor of
Mathematics and Statistics Mount Holyoke College Full review here [amazon]
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
7 of 45
Gil Strang Exalted Friend of the Matrix Professor of Mathematics at MIT since 1962 Many awardsincluding MAAHaimo Award forDistinguishedCollege orUniversity Teachingof Mathematics Rhodes Scholar Legend
More on Laplacian matrices graphs and othermadnesses here (Strangrsquos Wikipedia page is here
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
8 of 45
Admin
Potential paper products1 Outline
Papers to read1 ldquoThe Fundamental Theorem of Linear Algebrardquo [2]
2 ldquoToo Much Calculusrdquo [3]
Office hours 1000 to 1155 am WednesdaysFarrell Hall second floor Trinity Campus
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
9 of 45
Team MatrixologyWe may try out Slack Place for discussions about all things PoCS
including assignments and projects Once invited please sign up herehttpteam-matrixologyslackcom Very good Install Slack app on laptops tabletsphone Everyone will behave wonderfully
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
10 of 45
Grading breakdown
1 Levels (40) Ten one-week assignments Lowest assignment score will be dropped The last assignment cannot be dropped Each assignment will have a random bonus pointquestion which has nothing to do with linearalgebra
2 Challenge Levels (30) Three 75 minutes tests distributed throughout thecourse all of equal weighting
3 Final Boss Level (20) le Three hours of joyful celebration Thursday December 15 130 pm to 415 pm inPerkins 107
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
11 of 45
Grading breakdown
4 Mini-levels (10) Most meeting times will end with a 10 to 15minute mini-level There will be around 20 mini-levels
5 Homework (0)mdashProblems assigned online fromthe textbook Doing these exercises will be mostbeneficial and will increase happiness
6 General existencemdashit is extremely desirable thatstudents attend class and class presence will betaken into account if a grade is borderline
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
12 of 45
Questions are worth 3 points according to thefollowing scale 3 = correct or very nearly so 2 = acceptable but needs some revisions 1 = needs major revisions 0 = way off
Schedule The course will mainly cover chapters 2 through 6of the textbook (You should know all about Chapter 1)
Week (dates) Tuesday Thursday1 (830 and 901) = = + Level 12 (906 and 908) = = + Level 23 (913 and 915) = = + Level 34 (920 and 922) = and review Challenge Level 15 (927 and 929) Big picture Big picture + Level 46 (1004 and 1006) Big picture Big picture + Level 57 (1011 and 1013) Big picture Big picture + Level 68 (1018 and 1020) Big picture Challenge Level 29 (1025 and 1027) Normal equation Gram-Schmidt Process +
Level 710 (1101 and 1103) Eigenstuff Eigenstuff + Level 811 (1108 and 1110) Determinants Determinants + Level 912 (1115 and 1117) Eigenstuff textitChallenge Level 313 (1122 and 1124) Thanksgiving Thanksgiving14 (1129 and 1201) Positive Definite Matrices
+ Level 10SVD
15 (1206) SVD mdash
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
14 of 45
Important dates1 Classes run from Tuesday August 30 to Tuesday
December 62 AddDrop Audit PassNo Pass
deadlinemdashMonday September 123 Last day to withdrawmdashMonday October 31
(Sadness)4 Reading and Exam periodmdashSaturday December
10 to Friday December 16
More stuffDo check your zoo account for updates regarding thecourse
Academic assistance Anyone who requires assistancein any way (as per the ACCESS program or due toathletic endeavors) please see or contact me as soonas possible
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
15 of 45
More stuff
Being good people1 In class there will be no electronic gadgetry no cell
phones no beeping no text messaging etc Youreally just need your brain some paper and awriting implement here (okay and Matlab)
2 Second I encourage you to email me questionsideas comments etc about the class but requestthat you please do so in a respectful fashion
3 Finally as in all UVM classes Academic honestywill be expected and departures will be dealt withappropriately See httpwwwuvmeducses forguidelines
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
16 of 45
Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0
Computing Approximately 2 out of 10 questions perassignment will be Matlab based
Note for assignment problems written details ofcalculations will be required
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
4 of 45
Basics
Instructor Prof Peter Dodds Lecture room and meeting timesPerkins 107Tuesday and Thursday 1005 am to 1120 am Office Farrell Hall second floor Trinity Campus E-mail peterdoddsuvmedu Course websitehttpwwwuvmedu pdoddsteachingcourses2016-08UVM-122 Textbook ldquoIntroduction to Linear Algebrardquo (3rd or4th or 5th edition) by Gilbert Strang (published byWellesley-Cambridge Press)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
5 of 45
Our Textbook of Excellence
4th Edition V 3rd Edition V Unhelpful 2 ldquoIntroduction to Linear Algebrardquoby Gil Strang Textbook websitehttpmathmitedulinearalgebra MIT Open Courseware site for 1806(=Linear Algebra)httpocwmitedulinear-algebra-spring-2010
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
6 of 45
Yesness
Money quote from George Cobbrsquos review ofStrangrsquos bookDo you want a book written by a mathematician with alifetime experience using linear algebra to understandimportant authentic applied problems a formerpresident of the Society for Industrial and AppliedMathematics hellip
or do you want a book shaped mainly by the[a]esthetics of pure mathematicians with only a weaktheoretical connection to how linear algebra is used inthe natural and social sciences George Cobb Robert L Rooke Professor of
Mathematics and Statistics Mount Holyoke College Full review here [amazon]
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
7 of 45
Gil Strang Exalted Friend of the Matrix Professor of Mathematics at MIT since 1962 Many awardsincluding MAAHaimo Award forDistinguishedCollege orUniversity Teachingof Mathematics Rhodes Scholar Legend
More on Laplacian matrices graphs and othermadnesses here (Strangrsquos Wikipedia page is here
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
8 of 45
Admin
Potential paper products1 Outline
Papers to read1 ldquoThe Fundamental Theorem of Linear Algebrardquo [2]
2 ldquoToo Much Calculusrdquo [3]
Office hours 1000 to 1155 am WednesdaysFarrell Hall second floor Trinity Campus
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
9 of 45
Team MatrixologyWe may try out Slack Place for discussions about all things PoCS
including assignments and projects Once invited please sign up herehttpteam-matrixologyslackcom Very good Install Slack app on laptops tabletsphone Everyone will behave wonderfully
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
10 of 45
Grading breakdown
1 Levels (40) Ten one-week assignments Lowest assignment score will be dropped The last assignment cannot be dropped Each assignment will have a random bonus pointquestion which has nothing to do with linearalgebra
2 Challenge Levels (30) Three 75 minutes tests distributed throughout thecourse all of equal weighting
3 Final Boss Level (20) le Three hours of joyful celebration Thursday December 15 130 pm to 415 pm inPerkins 107
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
11 of 45
Grading breakdown
4 Mini-levels (10) Most meeting times will end with a 10 to 15minute mini-level There will be around 20 mini-levels
5 Homework (0)mdashProblems assigned online fromthe textbook Doing these exercises will be mostbeneficial and will increase happiness
6 General existencemdashit is extremely desirable thatstudents attend class and class presence will betaken into account if a grade is borderline
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
12 of 45
Questions are worth 3 points according to thefollowing scale 3 = correct or very nearly so 2 = acceptable but needs some revisions 1 = needs major revisions 0 = way off
Schedule The course will mainly cover chapters 2 through 6of the textbook (You should know all about Chapter 1)
Week (dates) Tuesday Thursday1 (830 and 901) = = + Level 12 (906 and 908) = = + Level 23 (913 and 915) = = + Level 34 (920 and 922) = and review Challenge Level 15 (927 and 929) Big picture Big picture + Level 46 (1004 and 1006) Big picture Big picture + Level 57 (1011 and 1013) Big picture Big picture + Level 68 (1018 and 1020) Big picture Challenge Level 29 (1025 and 1027) Normal equation Gram-Schmidt Process +
Level 710 (1101 and 1103) Eigenstuff Eigenstuff + Level 811 (1108 and 1110) Determinants Determinants + Level 912 (1115 and 1117) Eigenstuff textitChallenge Level 313 (1122 and 1124) Thanksgiving Thanksgiving14 (1129 and 1201) Positive Definite Matrices
+ Level 10SVD
15 (1206) SVD mdash
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
14 of 45
Important dates1 Classes run from Tuesday August 30 to Tuesday
December 62 AddDrop Audit PassNo Pass
deadlinemdashMonday September 123 Last day to withdrawmdashMonday October 31
(Sadness)4 Reading and Exam periodmdashSaturday December
10 to Friday December 16
More stuffDo check your zoo account for updates regarding thecourse
Academic assistance Anyone who requires assistancein any way (as per the ACCESS program or due toathletic endeavors) please see or contact me as soonas possible
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
15 of 45
More stuff
Being good people1 In class there will be no electronic gadgetry no cell
phones no beeping no text messaging etc Youreally just need your brain some paper and awriting implement here (okay and Matlab)
2 Second I encourage you to email me questionsideas comments etc about the class but requestthat you please do so in a respectful fashion
3 Finally as in all UVM classes Academic honestywill be expected and departures will be dealt withappropriately See httpwwwuvmeducses forguidelines
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
16 of 45
Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0
Computing Approximately 2 out of 10 questions perassignment will be Matlab based
Note for assignment problems written details ofcalculations will be required
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
5 of 45
Our Textbook of Excellence
4th Edition V 3rd Edition V Unhelpful 2 ldquoIntroduction to Linear Algebrardquoby Gil Strang Textbook websitehttpmathmitedulinearalgebra MIT Open Courseware site for 1806(=Linear Algebra)httpocwmitedulinear-algebra-spring-2010
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
6 of 45
Yesness
Money quote from George Cobbrsquos review ofStrangrsquos bookDo you want a book written by a mathematician with alifetime experience using linear algebra to understandimportant authentic applied problems a formerpresident of the Society for Industrial and AppliedMathematics hellip
or do you want a book shaped mainly by the[a]esthetics of pure mathematicians with only a weaktheoretical connection to how linear algebra is used inthe natural and social sciences George Cobb Robert L Rooke Professor of
Mathematics and Statistics Mount Holyoke College Full review here [amazon]
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
7 of 45
Gil Strang Exalted Friend of the Matrix Professor of Mathematics at MIT since 1962 Many awardsincluding MAAHaimo Award forDistinguishedCollege orUniversity Teachingof Mathematics Rhodes Scholar Legend
More on Laplacian matrices graphs and othermadnesses here (Strangrsquos Wikipedia page is here
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
8 of 45
Admin
Potential paper products1 Outline
Papers to read1 ldquoThe Fundamental Theorem of Linear Algebrardquo [2]
2 ldquoToo Much Calculusrdquo [3]
Office hours 1000 to 1155 am WednesdaysFarrell Hall second floor Trinity Campus
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
9 of 45
Team MatrixologyWe may try out Slack Place for discussions about all things PoCS
including assignments and projects Once invited please sign up herehttpteam-matrixologyslackcom Very good Install Slack app on laptops tabletsphone Everyone will behave wonderfully
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
10 of 45
Grading breakdown
1 Levels (40) Ten one-week assignments Lowest assignment score will be dropped The last assignment cannot be dropped Each assignment will have a random bonus pointquestion which has nothing to do with linearalgebra
2 Challenge Levels (30) Three 75 minutes tests distributed throughout thecourse all of equal weighting
3 Final Boss Level (20) le Three hours of joyful celebration Thursday December 15 130 pm to 415 pm inPerkins 107
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
11 of 45
Grading breakdown
4 Mini-levels (10) Most meeting times will end with a 10 to 15minute mini-level There will be around 20 mini-levels
5 Homework (0)mdashProblems assigned online fromthe textbook Doing these exercises will be mostbeneficial and will increase happiness
6 General existencemdashit is extremely desirable thatstudents attend class and class presence will betaken into account if a grade is borderline
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
12 of 45
Questions are worth 3 points according to thefollowing scale 3 = correct or very nearly so 2 = acceptable but needs some revisions 1 = needs major revisions 0 = way off
Schedule The course will mainly cover chapters 2 through 6of the textbook (You should know all about Chapter 1)
Week (dates) Tuesday Thursday1 (830 and 901) = = + Level 12 (906 and 908) = = + Level 23 (913 and 915) = = + Level 34 (920 and 922) = and review Challenge Level 15 (927 and 929) Big picture Big picture + Level 46 (1004 and 1006) Big picture Big picture + Level 57 (1011 and 1013) Big picture Big picture + Level 68 (1018 and 1020) Big picture Challenge Level 29 (1025 and 1027) Normal equation Gram-Schmidt Process +
Level 710 (1101 and 1103) Eigenstuff Eigenstuff + Level 811 (1108 and 1110) Determinants Determinants + Level 912 (1115 and 1117) Eigenstuff textitChallenge Level 313 (1122 and 1124) Thanksgiving Thanksgiving14 (1129 and 1201) Positive Definite Matrices
+ Level 10SVD
15 (1206) SVD mdash
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
14 of 45
Important dates1 Classes run from Tuesday August 30 to Tuesday
December 62 AddDrop Audit PassNo Pass
deadlinemdashMonday September 123 Last day to withdrawmdashMonday October 31
(Sadness)4 Reading and Exam periodmdashSaturday December
10 to Friday December 16
More stuffDo check your zoo account for updates regarding thecourse
Academic assistance Anyone who requires assistancein any way (as per the ACCESS program or due toathletic endeavors) please see or contact me as soonas possible
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
15 of 45
More stuff
Being good people1 In class there will be no electronic gadgetry no cell
phones no beeping no text messaging etc Youreally just need your brain some paper and awriting implement here (okay and Matlab)
2 Second I encourage you to email me questionsideas comments etc about the class but requestthat you please do so in a respectful fashion
3 Finally as in all UVM classes Academic honestywill be expected and departures will be dealt withappropriately See httpwwwuvmeducses forguidelines
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
16 of 45
Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0
Computing Approximately 2 out of 10 questions perassignment will be Matlab based
Note for assignment problems written details ofcalculations will be required
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
6 of 45
Yesness
Money quote from George Cobbrsquos review ofStrangrsquos bookDo you want a book written by a mathematician with alifetime experience using linear algebra to understandimportant authentic applied problems a formerpresident of the Society for Industrial and AppliedMathematics hellip
or do you want a book shaped mainly by the[a]esthetics of pure mathematicians with only a weaktheoretical connection to how linear algebra is used inthe natural and social sciences George Cobb Robert L Rooke Professor of
Mathematics and Statistics Mount Holyoke College Full review here [amazon]
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
7 of 45
Gil Strang Exalted Friend of the Matrix Professor of Mathematics at MIT since 1962 Many awardsincluding MAAHaimo Award forDistinguishedCollege orUniversity Teachingof Mathematics Rhodes Scholar Legend
More on Laplacian matrices graphs and othermadnesses here (Strangrsquos Wikipedia page is here
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
8 of 45
Admin
Potential paper products1 Outline
Papers to read1 ldquoThe Fundamental Theorem of Linear Algebrardquo [2]
2 ldquoToo Much Calculusrdquo [3]
Office hours 1000 to 1155 am WednesdaysFarrell Hall second floor Trinity Campus
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
9 of 45
Team MatrixologyWe may try out Slack Place for discussions about all things PoCS
including assignments and projects Once invited please sign up herehttpteam-matrixologyslackcom Very good Install Slack app on laptops tabletsphone Everyone will behave wonderfully
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
10 of 45
Grading breakdown
1 Levels (40) Ten one-week assignments Lowest assignment score will be dropped The last assignment cannot be dropped Each assignment will have a random bonus pointquestion which has nothing to do with linearalgebra
2 Challenge Levels (30) Three 75 minutes tests distributed throughout thecourse all of equal weighting
3 Final Boss Level (20) le Three hours of joyful celebration Thursday December 15 130 pm to 415 pm inPerkins 107
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
11 of 45
Grading breakdown
4 Mini-levels (10) Most meeting times will end with a 10 to 15minute mini-level There will be around 20 mini-levels
5 Homework (0)mdashProblems assigned online fromthe textbook Doing these exercises will be mostbeneficial and will increase happiness
6 General existencemdashit is extremely desirable thatstudents attend class and class presence will betaken into account if a grade is borderline
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
12 of 45
Questions are worth 3 points according to thefollowing scale 3 = correct or very nearly so 2 = acceptable but needs some revisions 1 = needs major revisions 0 = way off
Schedule The course will mainly cover chapters 2 through 6of the textbook (You should know all about Chapter 1)
Week (dates) Tuesday Thursday1 (830 and 901) = = + Level 12 (906 and 908) = = + Level 23 (913 and 915) = = + Level 34 (920 and 922) = and review Challenge Level 15 (927 and 929) Big picture Big picture + Level 46 (1004 and 1006) Big picture Big picture + Level 57 (1011 and 1013) Big picture Big picture + Level 68 (1018 and 1020) Big picture Challenge Level 29 (1025 and 1027) Normal equation Gram-Schmidt Process +
Level 710 (1101 and 1103) Eigenstuff Eigenstuff + Level 811 (1108 and 1110) Determinants Determinants + Level 912 (1115 and 1117) Eigenstuff textitChallenge Level 313 (1122 and 1124) Thanksgiving Thanksgiving14 (1129 and 1201) Positive Definite Matrices
+ Level 10SVD
15 (1206) SVD mdash
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
14 of 45
Important dates1 Classes run from Tuesday August 30 to Tuesday
December 62 AddDrop Audit PassNo Pass
deadlinemdashMonday September 123 Last day to withdrawmdashMonday October 31
(Sadness)4 Reading and Exam periodmdashSaturday December
10 to Friday December 16
More stuffDo check your zoo account for updates regarding thecourse
Academic assistance Anyone who requires assistancein any way (as per the ACCESS program or due toathletic endeavors) please see or contact me as soonas possible
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
15 of 45
More stuff
Being good people1 In class there will be no electronic gadgetry no cell
phones no beeping no text messaging etc Youreally just need your brain some paper and awriting implement here (okay and Matlab)
2 Second I encourage you to email me questionsideas comments etc about the class but requestthat you please do so in a respectful fashion
3 Finally as in all UVM classes Academic honestywill be expected and departures will be dealt withappropriately See httpwwwuvmeducses forguidelines
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
16 of 45
Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0
Computing Approximately 2 out of 10 questions perassignment will be Matlab based
Note for assignment problems written details ofcalculations will be required
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
7 of 45
Gil Strang Exalted Friend of the Matrix Professor of Mathematics at MIT since 1962 Many awardsincluding MAAHaimo Award forDistinguishedCollege orUniversity Teachingof Mathematics Rhodes Scholar Legend
More on Laplacian matrices graphs and othermadnesses here (Strangrsquos Wikipedia page is here
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
8 of 45
Admin
Potential paper products1 Outline
Papers to read1 ldquoThe Fundamental Theorem of Linear Algebrardquo [2]
2 ldquoToo Much Calculusrdquo [3]
Office hours 1000 to 1155 am WednesdaysFarrell Hall second floor Trinity Campus
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
9 of 45
Team MatrixologyWe may try out Slack Place for discussions about all things PoCS
including assignments and projects Once invited please sign up herehttpteam-matrixologyslackcom Very good Install Slack app on laptops tabletsphone Everyone will behave wonderfully
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
10 of 45
Grading breakdown
1 Levels (40) Ten one-week assignments Lowest assignment score will be dropped The last assignment cannot be dropped Each assignment will have a random bonus pointquestion which has nothing to do with linearalgebra
2 Challenge Levels (30) Three 75 minutes tests distributed throughout thecourse all of equal weighting
3 Final Boss Level (20) le Three hours of joyful celebration Thursday December 15 130 pm to 415 pm inPerkins 107
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
11 of 45
Grading breakdown
4 Mini-levels (10) Most meeting times will end with a 10 to 15minute mini-level There will be around 20 mini-levels
5 Homework (0)mdashProblems assigned online fromthe textbook Doing these exercises will be mostbeneficial and will increase happiness
6 General existencemdashit is extremely desirable thatstudents attend class and class presence will betaken into account if a grade is borderline
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
12 of 45
Questions are worth 3 points according to thefollowing scale 3 = correct or very nearly so 2 = acceptable but needs some revisions 1 = needs major revisions 0 = way off
Schedule The course will mainly cover chapters 2 through 6of the textbook (You should know all about Chapter 1)
Week (dates) Tuesday Thursday1 (830 and 901) = = + Level 12 (906 and 908) = = + Level 23 (913 and 915) = = + Level 34 (920 and 922) = and review Challenge Level 15 (927 and 929) Big picture Big picture + Level 46 (1004 and 1006) Big picture Big picture + Level 57 (1011 and 1013) Big picture Big picture + Level 68 (1018 and 1020) Big picture Challenge Level 29 (1025 and 1027) Normal equation Gram-Schmidt Process +
Level 710 (1101 and 1103) Eigenstuff Eigenstuff + Level 811 (1108 and 1110) Determinants Determinants + Level 912 (1115 and 1117) Eigenstuff textitChallenge Level 313 (1122 and 1124) Thanksgiving Thanksgiving14 (1129 and 1201) Positive Definite Matrices
+ Level 10SVD
15 (1206) SVD mdash
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
14 of 45
Important dates1 Classes run from Tuesday August 30 to Tuesday
December 62 AddDrop Audit PassNo Pass
deadlinemdashMonday September 123 Last day to withdrawmdashMonday October 31
(Sadness)4 Reading and Exam periodmdashSaturday December
10 to Friday December 16
More stuffDo check your zoo account for updates regarding thecourse
Academic assistance Anyone who requires assistancein any way (as per the ACCESS program or due toathletic endeavors) please see or contact me as soonas possible
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
15 of 45
More stuff
Being good people1 In class there will be no electronic gadgetry no cell
phones no beeping no text messaging etc Youreally just need your brain some paper and awriting implement here (okay and Matlab)
2 Second I encourage you to email me questionsideas comments etc about the class but requestthat you please do so in a respectful fashion
3 Finally as in all UVM classes Academic honestywill be expected and departures will be dealt withappropriately See httpwwwuvmeducses forguidelines
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
16 of 45
Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0
Computing Approximately 2 out of 10 questions perassignment will be Matlab based
Note for assignment problems written details ofcalculations will be required
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
8 of 45
Admin
Potential paper products1 Outline
Papers to read1 ldquoThe Fundamental Theorem of Linear Algebrardquo [2]
2 ldquoToo Much Calculusrdquo [3]
Office hours 1000 to 1155 am WednesdaysFarrell Hall second floor Trinity Campus
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
9 of 45
Team MatrixologyWe may try out Slack Place for discussions about all things PoCS
including assignments and projects Once invited please sign up herehttpteam-matrixologyslackcom Very good Install Slack app on laptops tabletsphone Everyone will behave wonderfully
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
10 of 45
Grading breakdown
1 Levels (40) Ten one-week assignments Lowest assignment score will be dropped The last assignment cannot be dropped Each assignment will have a random bonus pointquestion which has nothing to do with linearalgebra
2 Challenge Levels (30) Three 75 minutes tests distributed throughout thecourse all of equal weighting
3 Final Boss Level (20) le Three hours of joyful celebration Thursday December 15 130 pm to 415 pm inPerkins 107
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
11 of 45
Grading breakdown
4 Mini-levels (10) Most meeting times will end with a 10 to 15minute mini-level There will be around 20 mini-levels
5 Homework (0)mdashProblems assigned online fromthe textbook Doing these exercises will be mostbeneficial and will increase happiness
6 General existencemdashit is extremely desirable thatstudents attend class and class presence will betaken into account if a grade is borderline
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
12 of 45
Questions are worth 3 points according to thefollowing scale 3 = correct or very nearly so 2 = acceptable but needs some revisions 1 = needs major revisions 0 = way off
Schedule The course will mainly cover chapters 2 through 6of the textbook (You should know all about Chapter 1)
Week (dates) Tuesday Thursday1 (830 and 901) = = + Level 12 (906 and 908) = = + Level 23 (913 and 915) = = + Level 34 (920 and 922) = and review Challenge Level 15 (927 and 929) Big picture Big picture + Level 46 (1004 and 1006) Big picture Big picture + Level 57 (1011 and 1013) Big picture Big picture + Level 68 (1018 and 1020) Big picture Challenge Level 29 (1025 and 1027) Normal equation Gram-Schmidt Process +
Level 710 (1101 and 1103) Eigenstuff Eigenstuff + Level 811 (1108 and 1110) Determinants Determinants + Level 912 (1115 and 1117) Eigenstuff textitChallenge Level 313 (1122 and 1124) Thanksgiving Thanksgiving14 (1129 and 1201) Positive Definite Matrices
+ Level 10SVD
15 (1206) SVD mdash
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
14 of 45
Important dates1 Classes run from Tuesday August 30 to Tuesday
December 62 AddDrop Audit PassNo Pass
deadlinemdashMonday September 123 Last day to withdrawmdashMonday October 31
(Sadness)4 Reading and Exam periodmdashSaturday December
10 to Friday December 16
More stuffDo check your zoo account for updates regarding thecourse
Academic assistance Anyone who requires assistancein any way (as per the ACCESS program or due toathletic endeavors) please see or contact me as soonas possible
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
15 of 45
More stuff
Being good people1 In class there will be no electronic gadgetry no cell
phones no beeping no text messaging etc Youreally just need your brain some paper and awriting implement here (okay and Matlab)
2 Second I encourage you to email me questionsideas comments etc about the class but requestthat you please do so in a respectful fashion
3 Finally as in all UVM classes Academic honestywill be expected and departures will be dealt withappropriately See httpwwwuvmeducses forguidelines
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
16 of 45
Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0
Computing Approximately 2 out of 10 questions perassignment will be Matlab based
Note for assignment problems written details ofcalculations will be required
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
9 of 45
Team MatrixologyWe may try out Slack Place for discussions about all things PoCS
including assignments and projects Once invited please sign up herehttpteam-matrixologyslackcom Very good Install Slack app on laptops tabletsphone Everyone will behave wonderfully
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
10 of 45
Grading breakdown
1 Levels (40) Ten one-week assignments Lowest assignment score will be dropped The last assignment cannot be dropped Each assignment will have a random bonus pointquestion which has nothing to do with linearalgebra
2 Challenge Levels (30) Three 75 minutes tests distributed throughout thecourse all of equal weighting
3 Final Boss Level (20) le Three hours of joyful celebration Thursday December 15 130 pm to 415 pm inPerkins 107
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
11 of 45
Grading breakdown
4 Mini-levels (10) Most meeting times will end with a 10 to 15minute mini-level There will be around 20 mini-levels
5 Homework (0)mdashProblems assigned online fromthe textbook Doing these exercises will be mostbeneficial and will increase happiness
6 General existencemdashit is extremely desirable thatstudents attend class and class presence will betaken into account if a grade is borderline
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
12 of 45
Questions are worth 3 points according to thefollowing scale 3 = correct or very nearly so 2 = acceptable but needs some revisions 1 = needs major revisions 0 = way off
Schedule The course will mainly cover chapters 2 through 6of the textbook (You should know all about Chapter 1)
Week (dates) Tuesday Thursday1 (830 and 901) = = + Level 12 (906 and 908) = = + Level 23 (913 and 915) = = + Level 34 (920 and 922) = and review Challenge Level 15 (927 and 929) Big picture Big picture + Level 46 (1004 and 1006) Big picture Big picture + Level 57 (1011 and 1013) Big picture Big picture + Level 68 (1018 and 1020) Big picture Challenge Level 29 (1025 and 1027) Normal equation Gram-Schmidt Process +
Level 710 (1101 and 1103) Eigenstuff Eigenstuff + Level 811 (1108 and 1110) Determinants Determinants + Level 912 (1115 and 1117) Eigenstuff textitChallenge Level 313 (1122 and 1124) Thanksgiving Thanksgiving14 (1129 and 1201) Positive Definite Matrices
+ Level 10SVD
15 (1206) SVD mdash
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
14 of 45
Important dates1 Classes run from Tuesday August 30 to Tuesday
December 62 AddDrop Audit PassNo Pass
deadlinemdashMonday September 123 Last day to withdrawmdashMonday October 31
(Sadness)4 Reading and Exam periodmdashSaturday December
10 to Friday December 16
More stuffDo check your zoo account for updates regarding thecourse
Academic assistance Anyone who requires assistancein any way (as per the ACCESS program or due toathletic endeavors) please see or contact me as soonas possible
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
15 of 45
More stuff
Being good people1 In class there will be no electronic gadgetry no cell
phones no beeping no text messaging etc Youreally just need your brain some paper and awriting implement here (okay and Matlab)
2 Second I encourage you to email me questionsideas comments etc about the class but requestthat you please do so in a respectful fashion
3 Finally as in all UVM classes Academic honestywill be expected and departures will be dealt withappropriately See httpwwwuvmeducses forguidelines
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
16 of 45
Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0
Computing Approximately 2 out of 10 questions perassignment will be Matlab based
Note for assignment problems written details ofcalculations will be required
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
10 of 45
Grading breakdown
1 Levels (40) Ten one-week assignments Lowest assignment score will be dropped The last assignment cannot be dropped Each assignment will have a random bonus pointquestion which has nothing to do with linearalgebra
2 Challenge Levels (30) Three 75 minutes tests distributed throughout thecourse all of equal weighting
3 Final Boss Level (20) le Three hours of joyful celebration Thursday December 15 130 pm to 415 pm inPerkins 107
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
11 of 45
Grading breakdown
4 Mini-levels (10) Most meeting times will end with a 10 to 15minute mini-level There will be around 20 mini-levels
5 Homework (0)mdashProblems assigned online fromthe textbook Doing these exercises will be mostbeneficial and will increase happiness
6 General existencemdashit is extremely desirable thatstudents attend class and class presence will betaken into account if a grade is borderline
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
12 of 45
Questions are worth 3 points according to thefollowing scale 3 = correct or very nearly so 2 = acceptable but needs some revisions 1 = needs major revisions 0 = way off
Schedule The course will mainly cover chapters 2 through 6of the textbook (You should know all about Chapter 1)
Week (dates) Tuesday Thursday1 (830 and 901) = = + Level 12 (906 and 908) = = + Level 23 (913 and 915) = = + Level 34 (920 and 922) = and review Challenge Level 15 (927 and 929) Big picture Big picture + Level 46 (1004 and 1006) Big picture Big picture + Level 57 (1011 and 1013) Big picture Big picture + Level 68 (1018 and 1020) Big picture Challenge Level 29 (1025 and 1027) Normal equation Gram-Schmidt Process +
Level 710 (1101 and 1103) Eigenstuff Eigenstuff + Level 811 (1108 and 1110) Determinants Determinants + Level 912 (1115 and 1117) Eigenstuff textitChallenge Level 313 (1122 and 1124) Thanksgiving Thanksgiving14 (1129 and 1201) Positive Definite Matrices
+ Level 10SVD
15 (1206) SVD mdash
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
14 of 45
Important dates1 Classes run from Tuesday August 30 to Tuesday
December 62 AddDrop Audit PassNo Pass
deadlinemdashMonday September 123 Last day to withdrawmdashMonday October 31
(Sadness)4 Reading and Exam periodmdashSaturday December
10 to Friday December 16
More stuffDo check your zoo account for updates regarding thecourse
Academic assistance Anyone who requires assistancein any way (as per the ACCESS program or due toathletic endeavors) please see or contact me as soonas possible
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
15 of 45
More stuff
Being good people1 In class there will be no electronic gadgetry no cell
phones no beeping no text messaging etc Youreally just need your brain some paper and awriting implement here (okay and Matlab)
2 Second I encourage you to email me questionsideas comments etc about the class but requestthat you please do so in a respectful fashion
3 Finally as in all UVM classes Academic honestywill be expected and departures will be dealt withappropriately See httpwwwuvmeducses forguidelines
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
16 of 45
Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0
Computing Approximately 2 out of 10 questions perassignment will be Matlab based
Note for assignment problems written details ofcalculations will be required
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
11 of 45
Grading breakdown
4 Mini-levels (10) Most meeting times will end with a 10 to 15minute mini-level There will be around 20 mini-levels
5 Homework (0)mdashProblems assigned online fromthe textbook Doing these exercises will be mostbeneficial and will increase happiness
6 General existencemdashit is extremely desirable thatstudents attend class and class presence will betaken into account if a grade is borderline
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
12 of 45
Questions are worth 3 points according to thefollowing scale 3 = correct or very nearly so 2 = acceptable but needs some revisions 1 = needs major revisions 0 = way off
Schedule The course will mainly cover chapters 2 through 6of the textbook (You should know all about Chapter 1)
Week (dates) Tuesday Thursday1 (830 and 901) = = + Level 12 (906 and 908) = = + Level 23 (913 and 915) = = + Level 34 (920 and 922) = and review Challenge Level 15 (927 and 929) Big picture Big picture + Level 46 (1004 and 1006) Big picture Big picture + Level 57 (1011 and 1013) Big picture Big picture + Level 68 (1018 and 1020) Big picture Challenge Level 29 (1025 and 1027) Normal equation Gram-Schmidt Process +
Level 710 (1101 and 1103) Eigenstuff Eigenstuff + Level 811 (1108 and 1110) Determinants Determinants + Level 912 (1115 and 1117) Eigenstuff textitChallenge Level 313 (1122 and 1124) Thanksgiving Thanksgiving14 (1129 and 1201) Positive Definite Matrices
+ Level 10SVD
15 (1206) SVD mdash
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
14 of 45
Important dates1 Classes run from Tuesday August 30 to Tuesday
December 62 AddDrop Audit PassNo Pass
deadlinemdashMonday September 123 Last day to withdrawmdashMonday October 31
(Sadness)4 Reading and Exam periodmdashSaturday December
10 to Friday December 16
More stuffDo check your zoo account for updates regarding thecourse
Academic assistance Anyone who requires assistancein any way (as per the ACCESS program or due toathletic endeavors) please see or contact me as soonas possible
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
15 of 45
More stuff
Being good people1 In class there will be no electronic gadgetry no cell
phones no beeping no text messaging etc Youreally just need your brain some paper and awriting implement here (okay and Matlab)
2 Second I encourage you to email me questionsideas comments etc about the class but requestthat you please do so in a respectful fashion
3 Finally as in all UVM classes Academic honestywill be expected and departures will be dealt withappropriately See httpwwwuvmeducses forguidelines
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
16 of 45
Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0
Computing Approximately 2 out of 10 questions perassignment will be Matlab based
Note for assignment problems written details ofcalculations will be required
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
12 of 45
Questions are worth 3 points according to thefollowing scale 3 = correct or very nearly so 2 = acceptable but needs some revisions 1 = needs major revisions 0 = way off
Schedule The course will mainly cover chapters 2 through 6of the textbook (You should know all about Chapter 1)
Week (dates) Tuesday Thursday1 (830 and 901) = = + Level 12 (906 and 908) = = + Level 23 (913 and 915) = = + Level 34 (920 and 922) = and review Challenge Level 15 (927 and 929) Big picture Big picture + Level 46 (1004 and 1006) Big picture Big picture + Level 57 (1011 and 1013) Big picture Big picture + Level 68 (1018 and 1020) Big picture Challenge Level 29 (1025 and 1027) Normal equation Gram-Schmidt Process +
Level 710 (1101 and 1103) Eigenstuff Eigenstuff + Level 811 (1108 and 1110) Determinants Determinants + Level 912 (1115 and 1117) Eigenstuff textitChallenge Level 313 (1122 and 1124) Thanksgiving Thanksgiving14 (1129 and 1201) Positive Definite Matrices
+ Level 10SVD
15 (1206) SVD mdash
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
14 of 45
Important dates1 Classes run from Tuesday August 30 to Tuesday
December 62 AddDrop Audit PassNo Pass
deadlinemdashMonday September 123 Last day to withdrawmdashMonday October 31
(Sadness)4 Reading and Exam periodmdashSaturday December
10 to Friday December 16
More stuffDo check your zoo account for updates regarding thecourse
Academic assistance Anyone who requires assistancein any way (as per the ACCESS program or due toathletic endeavors) please see or contact me as soonas possible
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
15 of 45
More stuff
Being good people1 In class there will be no electronic gadgetry no cell
phones no beeping no text messaging etc Youreally just need your brain some paper and awriting implement here (okay and Matlab)
2 Second I encourage you to email me questionsideas comments etc about the class but requestthat you please do so in a respectful fashion
3 Finally as in all UVM classes Academic honestywill be expected and departures will be dealt withappropriately See httpwwwuvmeducses forguidelines
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
16 of 45
Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0
Computing Approximately 2 out of 10 questions perassignment will be Matlab based
Note for assignment problems written details ofcalculations will be required
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Schedule The course will mainly cover chapters 2 through 6of the textbook (You should know all about Chapter 1)
Week (dates) Tuesday Thursday1 (830 and 901) = = + Level 12 (906 and 908) = = + Level 23 (913 and 915) = = + Level 34 (920 and 922) = and review Challenge Level 15 (927 and 929) Big picture Big picture + Level 46 (1004 and 1006) Big picture Big picture + Level 57 (1011 and 1013) Big picture Big picture + Level 68 (1018 and 1020) Big picture Challenge Level 29 (1025 and 1027) Normal equation Gram-Schmidt Process +
Level 710 (1101 and 1103) Eigenstuff Eigenstuff + Level 811 (1108 and 1110) Determinants Determinants + Level 912 (1115 and 1117) Eigenstuff textitChallenge Level 313 (1122 and 1124) Thanksgiving Thanksgiving14 (1129 and 1201) Positive Definite Matrices
+ Level 10SVD
15 (1206) SVD mdash
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
14 of 45
Important dates1 Classes run from Tuesday August 30 to Tuesday
December 62 AddDrop Audit PassNo Pass
deadlinemdashMonday September 123 Last day to withdrawmdashMonday October 31
(Sadness)4 Reading and Exam periodmdashSaturday December
10 to Friday December 16
More stuffDo check your zoo account for updates regarding thecourse
Academic assistance Anyone who requires assistancein any way (as per the ACCESS program or due toathletic endeavors) please see or contact me as soonas possible
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
15 of 45
More stuff
Being good people1 In class there will be no electronic gadgetry no cell
phones no beeping no text messaging etc Youreally just need your brain some paper and awriting implement here (okay and Matlab)
2 Second I encourage you to email me questionsideas comments etc about the class but requestthat you please do so in a respectful fashion
3 Finally as in all UVM classes Academic honestywill be expected and departures will be dealt withappropriately See httpwwwuvmeducses forguidelines
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
16 of 45
Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0
Computing Approximately 2 out of 10 questions perassignment will be Matlab based
Note for assignment problems written details ofcalculations will be required
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
14 of 45
Important dates1 Classes run from Tuesday August 30 to Tuesday
December 62 AddDrop Audit PassNo Pass
deadlinemdashMonday September 123 Last day to withdrawmdashMonday October 31
(Sadness)4 Reading and Exam periodmdashSaturday December
10 to Friday December 16
More stuffDo check your zoo account for updates regarding thecourse
Academic assistance Anyone who requires assistancein any way (as per the ACCESS program or due toathletic endeavors) please see or contact me as soonas possible
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
15 of 45
More stuff
Being good people1 In class there will be no electronic gadgetry no cell
phones no beeping no text messaging etc Youreally just need your brain some paper and awriting implement here (okay and Matlab)
2 Second I encourage you to email me questionsideas comments etc about the class but requestthat you please do so in a respectful fashion
3 Finally as in all UVM classes Academic honestywill be expected and departures will be dealt withappropriately See httpwwwuvmeducses forguidelines
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
16 of 45
Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0
Computing Approximately 2 out of 10 questions perassignment will be Matlab based
Note for assignment problems written details ofcalculations will be required
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
15 of 45
More stuff
Being good people1 In class there will be no electronic gadgetry no cell
phones no beeping no text messaging etc Youreally just need your brain some paper and awriting implement here (okay and Matlab)
2 Second I encourage you to email me questionsideas comments etc about the class but requestthat you please do so in a respectful fashion
3 Finally as in all UVM classes Academic honestywill be expected and departures will be dealt withappropriately See httpwwwuvmeducses forguidelines
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
16 of 45
Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0
Computing Approximately 2 out of 10 questions perassignment will be Matlab based
Note for assignment problems written details ofcalculations will be required
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
16 of 45
Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0
Computing Approximately 2 out of 10 questions perassignment will be Matlab based
Note for assignment problems written details ofcalculations will be required
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
17 of 45
Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems
Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)
Even more
If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
18 of 45
Why are we doing thisLinear Algebra is used in many fields to solveproblems
Engineering Computer Science Physics
Biology Ecology Economics Science of theSociotechnocene
Big exampleGooglersquos Pagerank
Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
19 of 45
Why are we doing this
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
20 of 45
You are now choosing the red pill
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
21 of 45
The Truth
Calculus is the Serpentrsquos Mathematics
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
22 of 45
The Platypus of Truth
Platypuses are masters of Linear Algebra
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
The Actual Truth
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
24 of 45
Matrices as gadgets
A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
25 of 45
Digital photographs are matrices
Usually three matrices RGB color model
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
27 of 45
Best fit line (least squares)
100
101
102
103
104
105
106
107
10minus2
10minus1
100
101
102
103
104
Mass (g)
Bas
al m
etab
olis
m (
W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil
From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
28 of 45
The many delights of Eigenthings
Using Linear Algebra wersquoll somehow connect
FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
29 of 45
This is a math course
httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
30 of 45
Three key problems of Linear Algebra
1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such
that = 3 Coupled linear differential equations
dd() = ()
Our focus will be largely on 1 partly on 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
31 of 45
Major course objective
To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture
Column Space
Rn
Null Space
~xn
~xr
A ~xn = ~0
A~xr = ~p
A~xlowast = ~p
d = n minus r
d = r
~xlowast = ~xr + ~xn
~p
~b
~e
Avi = σiui
Space
Left Null
Rm
~0
~0
d = m minus r
d = r
Row Space
v1
vr+1 vn
ur
u1
ur+1um
vr
What is going on here We have 24 episodes to findout
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
32 of 45
The fourfold ways of = case example big picture
solutions = = [ 1 00 1 ] 1 always
= gt [ 1 0 K10 1 K2 ] infin always
gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1
gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
33 of 45
Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to
make (as best we can)How can we disentangle an orchestrarsquos sound
Radiolabrsquos amazing pieceA 4-Track Mind
What about pictures waves signals
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
34 of 45
Is this your left nullspace
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
35 of 45
Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace
See also The Dunning-Kruger effect
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
36 of 45
Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
37 of 45
Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture
Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
38 of 45
Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both
equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra
Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
39 of 45
Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
40 of 45
Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions
Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions
(assuming neither 1 or 1 are 0)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
41 of 45
Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to
as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
42 of 45
Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]
is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
43 of 45
The Matrix Picture
Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
44 of 45
More Truth about Mathematics
The Colbert Report on Math (February 7 2006)
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-
Episode 124
Introduction
Exciting Admin
Importance
Usages
Key problems
Three ways oflooking
Colbert onEquations
References
45 of 45
References I
[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf
[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf
[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf
- Exciting Admin
- Importance
- Usages
- Key problems
- Three ways of looking
- Colbert on Equations
- References
-