Download - Poll - people.math.gatech.edu
![Page 1: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/1.jpg)
Poll
What does
✓1 10 1
◆do to this letter F?
Poll
Polls chaninndteams
HIM stints
shear
![Page 2: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/2.jpg)
Announcements Sep 28
• WeBWorK on Sections 2.7, 2.9, 3.1 due Thursday night
• Quiz on Section 2.7, 2.9, 3.1 Friday 8 am - 8 pm EDT
• My O�ce Hours Tue 11-12, Thu 1-2, and by appointment• TA O�ce Hours
I Umar Fri 4:20-5:20I Seokbin Wed 10:30-11:30I Manuel Mon 5-6I Pu-ting Thu 3-4I Juntao Thu 3-4
• Studio on Friday
• Second Midterm Friday Oct 16 8 am - 8 pm on §2.6-3.6 (not §2.8)• Tutoring: http://tutoring.gatech.edu/tutoring
• PLUS sessions: http://tutoring.gatech.edu/plus-sessions
• Math Lab: http://tutoring.gatech.edu/drop-tutoring-help-desks
• For general questions, post on Piazza
• Find a group to work with - let me know if you need help
• Counseling center: https://counseling.gatech.edu
Mneed to change check email
![Page 3: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/3.jpg)
Sections 3.1Matrix Transformations
Chapter3reframing
chaps1 2
in termsof
algebra
![Page 4: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/4.jpg)
From matrices to functions
Let A be an m⇥ n matrix.
We define a function
T : Rn ! Rm
T (v) = Av
This is called a matrix transformation.
The domain of T is Rn.
The co-domain of T is Rm.
The range of T is the set of outputs: Col(A)
This gives us another point of view of Ax = b
: linear system, matrix equation,
vector equation, linear transformation equation
Demo
rule with inputs outputs each inputhas onlyone output
4
functioninputs
possible outputs
solving Ax b is findinginput x with output
b
fort
![Page 5: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/5.jpg)
Why are we learning about matrix transformations?
Sample applications:
• Cryptography (Hill cypher)
• Computer graphics (Perspective projection is a linear map!)
• Aerospace (Control systems - input/output)
• Biology
• Many more!
![Page 6: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/6.jpg)
Applications of Linear Algebra
Biology: In a population of rabbits...
• half of the new born rabbits survive their first year
• of those, half survive their second year
• the maximum life span is three years
• rabbits produce 0, 6, 8 rabbits in their first, second, and third years
If I know the population in 2016 (in terms of the number of first, second, and
third year rabbits), then what is the population in 2017?
These relations can be represented using a matrix.
0
@0 6 012 0 10 1
2 0
1
A
How does this relate to matrix transformations?
Demo
g1st2nd3rdyr
rabbits
in oneyear
88111 1
fX next year
input output
![Page 7: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/7.jpg)
Section 3.2One-to-one and onto transformations
![Page 8: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/8.jpg)
Section 3.2 Outline
• Learn the definitions of one-to-one and onto functions
• Determine if a given matrix transformation is one-to-one and/or onto
examplefCx x2notonto
l isnotintherange
In Calculus KeysOne to one Ontoeachhorizontalline
a horizontal linetest crossgraphs1 pt each horizontalline
eachinputhas oneoutput crossesgraph 31 atall possibleoutputs
outputare actualoutputs
one toone
codomain range
aNth
twoinputs
withsameoutput
![Page 9: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/9.jpg)
One-to-one
A matrix transformation T : Rn ! Rmis one-to-one if each b in Rm
is the
output for at most one v in Rn.
In other words: di↵erent inputs have di↵erent outputs.
Do not confuse this with the definition of a function, which says that for each
input x in Rnthere is at most one output b in Rm
.
![Page 10: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/10.jpg)
One-to-one
T : Rn ! Rmis one-to-one if each b in Rm
is the output for at most one v in
Rn.
Theorem. Suppose T : Rn ! Rmis a matrix transformation with matrix A.
Then the following are all equivalent:
• T is one-to-one
• the columns of A are linearly independent
• Ax = 0 has only the trivial solution
• A has a pivot in each column
• the range of T has dimension n
What can we say about the relative sizes of m and n if T is one-to-one?
Draw a picture of the range of a one-to-one matrix transformation R ! R3.
no one ITEM
gain.eadiariasidamo
Yahia initO
xyonlyOsdn
O QColla
O
m n tall or square not wide
i
![Page 11: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/11.jpg)
Onto
A matrix transformation T : Rn ! Rmis onto if the range of T equals the
codomain Rm, that is, each b in Rm
is the output for at least one input v in
Rm.
![Page 12: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/12.jpg)
Onto
T : Rn ! Rmis onto if the range of T equals the codomain Rm
, that is, each
b in Rmis the output for at least one input v in Rm
.
Theorem. Suppose T : Rn ! Rmis a matrix transformation with matrix A.
Then the following are all equivalent:
• T is onto
• the columns of A span Rm
• A has a pivot in each row
• Ax = b is consistent for all b in Rm
• the range of T has dimension m
What can we say about the relative sizes of m and n if T is onto?
Give an example of an onto matrix transformation R3 ! R.
Readaboutrobotarms in ILA
aerifying yangedgesL jgame I
OColla IRM Rowslinind
Msn wide or square nottall2
foie.im 9908
![Page 13: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/13.jpg)
One-to-one and Onto
Do the following give matrix transformations that are one-to-one? onto?
0
@1 0 70 1 20 0 9
1
A
0
@1 01 12 1
1
A✓
1 0 00 1 0
◆ ✓2 11 1
◆
Oo Yo0
one to oneKidnsind X
Vffnsind X
onto V X tallV X
By the way onto rows linind
![Page 14: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/14.jpg)
One-to-one and Onto
Which of the previously-studied matrix transformations of R2are one-to-one?
Onto?
✓0 11 0
◆reflection
✓1 00 0
◆projection
✓3 00 3
◆scaling
✓1 10 1
◆shear
✓cos ✓ � sin ✓sin ✓ cos ✓
◆rotation
One to one Onto
T Tabouty x
ontox an.is
ItFut
r
r r
v r
![Page 15: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/15.jpg)
Which are one to one / onto?
Which give one to one-to-one / onto matrix transforma-
tions?
✓1 1 00 1 1
◆ 0
@1 00 11 0
1
A✓
1 �1 2�2 2 �4
◆
Poll
Demo
Demo
Demo
fudai is oneto one
Like172
Inputs 22
not one to one sameoutput
botonetoone
Find 2 inputs with sameoutputforf
Inputs f il.fiOutput 0 Not one to onesame
g b
![Page 16: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/16.jpg)
Robot arm
Consider the robot arm example from the book.
There is a natural function f here (not a matrix transformation). The input is a
set of three angles and the co-domain is R2. Is this function one-to-one? Onto?
![Page 17: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/17.jpg)
Summary of Section 3.2
• T : Rn ! Rmis one-to-one if each b in Rm
is the output for at most one
v in Rn.
• Theorem. Suppose T : Rn ! Rmis a matrix transformation with matrix
A. Then the following are all equivalent:
I T is one-to-oneI the columns of A are linearly independentI Ax = 0 has only the trivial solutionI A has a pivot in each columnI the range has dimension n
• T : Rn ! Rmis onto if the range of T equals the codomain Rm
, that is,
each b in Rmis the output for at least one input v in Rm
.
• Theorem. Suppose T : Rn ! Rmis a matrix transformation with matrix
A. Then the following are all equivalent:
I T is ontoI the columns of A span Rm
I A has a pivot in each rowI Ax = b is consistent for all b in Rm.I the range of T has dimension m
![Page 18: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/18.jpg)
Typical exam questions
• True/False. It is possible for the matrix transformation for a 5⇥ 6 matrix
to be both one-to-one and onto.
• True/False. The matrix transformation T : R3 ! R3given by projection
to the yz-plane is onto.
• True/False. The matrix transformation T : R2 ! R2given by rotation by
⇡ is onto.
• Is there an onto matrix transformation R2 ! R3? If so, write one down, if
not explain why not.
• Is there an one-to-one matrix transformation R2 ! R3? If so, write one
down, if not explain why not.
![Page 19: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/19.jpg)
Section 3.3Linear Transformations
![Page 20: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/20.jpg)
Section 3.3 Outline
• Understand the definition of a linear transformation
• Linear transformations are the same as matrix transformations
• Find the matrix for a linear transformation
![Page 21: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/21.jpg)
Linear transformations
A function T : Rn ! Rm is a linear transformation if
• T (u+ v) = T (u) + T (v) for all u, v in Rn.
• T (cv) = cT (v) for all v in Rn and c in R.
First examples: matrix transformations.
Like 1in combos
bspaces
A utr Aut Av
A au c Au
![Page 22: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/22.jpg)
Linear transformations
A function T : Rn ! Rm is a linear transformation if
• T (u+ v) = T (u) + T (v) for all u, v in Rn.
• T (cv) = cT (v) for all v in Rn and c in R.
Notice that T (0) = 0. Why?
We have the standard basis vectors for Rn:
e1 = (1, 0, 0, . . . , 0)
e2 = (0, 1, 0, . . . , 0)
...
If we know T (e1), . . . , T (en), then we know every T (v). Why?
In engineering, this is called the principle of superposition.
![Page 23: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/23.jpg)
Linear transformations are matrix transformations
Theorem. Every linear transformation is a matrix transformation.
This means that for any linear transformation T : Rn ! Rm there is an m⇥ nmatrix A so that
T (v) = Av
for all v in Rn.
The matrix for a linear transformation is called the standard matrix.
![Page 24: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/24.jpg)
Linear transformations are matrix transformations
Theorem. Every linear transformation is a matrix transformation.
Given a linear transformation T : Rn ! Rm the standard matrix is:
A =
0
@| | |
T (e1) T (e2) · · · T (en)| | |
1
A
Why? Notice that Aei = T (ei) for all i. Then it follows from linearity thatT (v) = Av for all v.
![Page 25: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/25.jpg)
The identity
The identity linear transformation T : Rn ! Rn is
T (v) = v
What is the standard matrix?
This standard matrix is called In or I.
![Page 26: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/26.jpg)
Linear transformations are matrix transformations
Suppose T : R2 ! R3 is the function given by:
T
✓xy
◆=
0
@x+ y
yx� y
1
A
What is the standard matrix for T?
In fact, a function Rn ! Rm is linear exactly when the coordinates are linear(linear combinations of the variables, no constant terms).
![Page 27: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/27.jpg)
Linear transformations are matrix transformations
Find the standard matrix for the linear transformation of R2 that stretches by 2in the x-direction and 3 in the y-direction, and then reflects over the line y = x.
![Page 28: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/28.jpg)
Linear transformations are matrix transformations
Find the standard matrix for the linear transformation of R2 that projects ontothe y-axis and then rotates counterclockwise by ⇡/2.
![Page 29: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/29.jpg)
Linear transformations are matrix transformations
Find the standard matrix for the linear transformation of R3 that reflectsthrough the xy-plane and then projects onto the yz-plane.
![Page 30: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/30.jpg)
Discussion
Find a matrix that does this.
Discussion Question
![Page 31: Poll - people.math.gatech.edu](https://reader031.vdocument.in/reader031/viewer/2022012416/6171499b20a6a31570168137/html5/thumbnails/31.jpg)
Summary of 3.3
• A function T : Rn ! Rm is linear ifI T (u+ v) = T (u) + T (v) for all u, v in Rn.I T (cv) = cT (v) for all v 2 Rn and c in R.
• Theorem. Every linear transformation is a matrix transformation (andvice versa).
• The standard matrix for a linear transformation has its ith column equalto T (ei).