professor donna calhouncalhoun/teaching/...introduction to linear algebra, fourth edition gilbert...

Linear AlgebraLinear Algebra

Professor Donna Calhoun

Spring 2013

Office : MG 241AOffice Hours : Tuesday 11-12

(Math 301)

http://math.boisestate.edu/~calhoun/teaching/Math301_Spring2013

Thursday, January 24, 13



Textbook

Introduction to Linear Algebra, Fourth Edition

Gilbert Strang

$64 New (on Amazon)

Hardcover: 584 pagesPublisher: Wellesley Cambridge Press; 4 edition (February 10, 2009)Language: EnglishISBN-10: 0980232716ISBN-13: 978-0980232714


Algebra?Algebra?

3x

2� 5x+ 6 = 0

Solve for x : x

2� 93x+ 9

Simplify :

5x3 � 6x

4 + 3x3 � 7x+ 331Factoring polynomials :

px

� 3px

3 � 27,

e

ae

be

ln(a+b)Radicals, exponentials, logs

Power laws

(x

1/2y

3/2) �1

(x

�2� 3

y

5)

A

x+ 3+

Bx

(x+ 3)(x+ 1)

Partial fractions

1q1x

+ 1y

= 1 ! y =x

x� 1

Get y in terms of x :

x = �b± p

b

2� 4a

c2a

quadratic formula


Algebra?Linear equations



3x = 7

Solution is a scalar

x =7

3⇡ 2.33333



2x� y = 1

x+ y = 3

Solution is a point in the plane

3

3�1

x =4

3⇡ 1.333

y =5

3⇡ 1.666

3x = 7


x =7

3⇡ 2.33333



The equations are linear because no powers (other

than “0” or “1”) or products (“xy”) of x or y

appear

2x� y = 1

x+ y = 3

Solution is a point in the plane

3

3�1

x =4

3⇡ 1.333

y =5

3⇡ 1.666

3x = 7


x =7

3⇡ 2.33333


Linear equations

Each equation describes a line in the plane, or a plane in three-dimensional space. The solution (if one exists) is the intersection of the two lines or three planes.

2x+ y � 5z = 1

4x� 5y + z = 2

6x+ y + 2z = �5

x = ?, y = ?, z = ?

Solution is a point in three dimensional space

x

yz

3

3�1

Intersection of the three planes


What might we ask about the system?



2x+ y � 5z = 1

4x� 5y + z = 2

6x+ y + 2z = �5

x = ?, y = ?, z = ?

How do we know there is a solution?



2x+ y � 5z = 1

4x� 5y + z = 2

6x+ y + 2z = �5

x = ?, y = ?, z = ?


Parallel lines - no solution

In two dimensions



2x+ y � 5z = 1

4x� 5y + z = 2

6x+ y + 2z = �5

x = ?, y = ?, z = ?



In two dimensions

3

3�1

Exactly one solution



2x+ y � 5z = 1

4x� 5y + z = 2

6x+ y + 2z = �5

x = ?, y = ?, z = ?



In two dimensions

Co-linear - infinite number of solutions

3

3�1

Exactly one solution


How do we extend this idea?

12w + 4x+ 23y + 9z = 0

2u+ v + 5w � 2x+ 2y + 8z = 1

�5u+ v � 6w + 2x+ 4y � z = 6

8u� 4v � 5w � x� 7y = 7

11u+ 3v + 9x+ y + 9z = 11

3u� 2v � 8w � 15x+ 5y � 6z = 45

Does this system have a solution? How do we find the solution?


Other types of systems?



3x+ 8y = �4

an “underdetermined system” (not enough equations)

Question : How do we describe all of the solutions?



3x+ 8y = �4



Answer: A point (x,y) such that for any x,

y =3x

8+

1

2

All of the solutions lie on a line



3x+ 8y = �4



3x = 5

x = 1

an “overdetermined system” (too many equations)

Question : Is there a “best” solution?


y =3x

8+

1

2




3x+ 8y = �4



3x = 5

x = 1



Answer: Find solution “closest” to solutions to each equation.

bx =8

5not obvious!

3 5


y =3x

8+

1

2



More generally :


More generally :


u� v + 3x+ 8y + 5z = �4

3u+ 2v � x� y + z = 0

u+ 2v + x+ y + 5z = 10


More generally :



u� v + 3x+ 8y + 5z = �4

3u+ 2v � x� y + z = 0

u+ 2v + x+ y + 5z = 10


More generally :


Answer: ???


u� v + 3x+ 8y + 5z = �4

3u+ 2v � x� y + z = 0

u+ 2v + x+ y + 5z = 10


More generally :


Answer: ???


u� v + 3x+ 8y + 5z = �4

3u+ 2v � x� y + z = 0

u+ 2v + x+ y + 5z = 10


2x� y = 7

x+ 6y = 3

2x� 2y = 1

x+ y = 12

x� y = 3


More generally :


Answer: ???



u� v + 3x+ 8y + 5z = �4

3u+ 2v � x� y + z = 0

u+ 2v + x+ y + 5z = 10


2x� y = 7

x+ 6y = 3

2x� 2y = 1

x+ y = 12

x� y = 3


More generally :


Answer: ???


Answer: Find solution “closest” to solutions to each equation.


u� v + 3x+ 8y + 5z = �4

3u+ 2v � x� y + z = 0

u+ 2v + x+ y + 5z = 10


2x� y = 7

x+ 6y = 3

2x� 2y = 1

x+ y = 12

x� y = 3


Where do these equations come from?



Suppose you have several samples of a material that is made up of two distinct components. You (somehow) know the volume each component takes up in the sample, but not their respective densities . You can find the density of each component using the following model if you know the weight of each sample

⇢1, ⇢2⌫1, ⌫2

!




⇢1, ⇢2⌫1, ⌫2

!

Model : ⇢1⌫1 + ⇢2⌫2 = !




⇢1, ⇢2⌫1, ⌫2

!

Model : ⇢1⌫1 + ⇢2⌫2 = !

Data :

(4.12, 5.39, 1.09)

(4.13, 5.41, 1.20)

(3.91, 5.32, 1.11)

(3.89, 5.11, 1.02)

(2.11, 4.95, 1.05)

(⌫1, ⌫2,!)




⇢1, ⇢2⌫1, ⌫2

!

Model : ⇢1⌫1 + ⇢2⌫2 = !

Data :

(4.12, 5.39, 1.09)

(4.13, 5.41, 1.20)

(3.91, 5.32, 1.11)

(3.89, 5.11, 1.02)

(2.11, 4.95, 1.05)

(⌫1, ⌫2,!) Solve to get ⇢1, ⇢2

More equations than unknowns

4.12⇢1 + 5.39⇢2 = 1.09

4.13⇢1 + 5.41⇢2 = 1.20

3.91⇢1 + 5.32⇢2 = 1.11

3.89⇢1 + 5.11⇢2 = 1.02

4.21⇢1 + 4.95⇢2 = 1.05




⇢1, ⇢2⌫1, ⌫2

!

Model : ⇢1⌫1 + ⇢2⌫2 = !

Data :

(4.12, 5.39, 1.09)

(4.13, 5.41, 1.20)

(3.91, 5.32, 1.11)

(3.89, 5.11, 1.02)

(2.11, 4.95, 1.05)

(⌫1, ⌫2,!) Solve to get ⇢1, ⇢2

More equations than unknowns

4.12⇢1 + 5.39⇢2 = 1.09

4.13⇢1 + 5.41⇢2 = 1.20

3.91⇢1 + 5.32⇢2 = 1.11

3.89⇢1 + 5.11⇢2 = 1.02

4.21⇢1 + 4.95⇢2 = 1.05

“Best” solutionb⇢1 = 0.0371

b⇢2 = 0.1804


Systems with millions of unknowns

• 1.23 million “degrees of freedom” (DOF).

• Solves in 6.8 minutes on a desktop computer,

• Every point on the vehicle is an unknown.

DOF=number of equations


professor donna calhouncalhoun/teaching/...introduction to linear algebra, fourth edition gilbert...

Documents