Math21b Review to first midterm Spring 2007 1 Matrices 2 column - - PowerPoint PPT Presentation

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Math21b Review to first midterm Spring 2007 1 Matrices 2 column - - PowerPoint PPT Presentation

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Math21b Review to first midterm Spring 2007 1 Matrices 2 column picture x 1 x 2 Ax= v v v m 2 1 x m x 1 x m v v x 1 v + ... + = + m 2 1 3 row picture Ax= Example: A x =0, means x is perpendicular to row space.

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Math21b Review to first midterm Spring 2007

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Matrices

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column picture

Ax=

x1 x x1

+ =

x1 v

1

v

2

v

m

v

1

v

2 + ... +

x

m v m 2

xm

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row picture

Ax=

Example: A x =0, means x is perpendicular to row space. Example: A x =b, means b is the dot product of the k’th row with x.

k

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=

matrix multiplication

nxm

mxp

nxp

.

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Problem:

Can we multiply a 4 x 5 matrix A with a 5x4 matrix B? Is A B defined?

in other words:

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Matrix algebra:

With nxn matrices A,B,C,D,... one can work as with numbers

A+B = B+A a (A+B)=aA + aB A(B+C) = AB + AC A (B C) = (A B) C etc

except for two things in general:

A B = B A A

  • 1
  • 1 -1
  • 1

might not exist even for nonzero A

(A B) = B A

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True or False?

(A + B) ( A - B) = A - B

2 2

(1+A+A + A ) = (1-A ) (1-A)

2 3 4

  • 1

assuming (1-A) is invertible 1=In

A,B, arbitrary nxn matrices

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Row reduction

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Gauss-Jordan elimination

Scale a row Swap two rows Subtract row from

  • ther

row

S S S

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First blackboard problem

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Row reduce the X matrix

8 8 8 8 8 8 8 8 8

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Remember the Boston milkshake scare?

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Gotscha!

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You can no more escape

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Executed with water gun

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Warriors with their prey

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The milkshake as a matrix

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Lets row reduce it!

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The rref death of a milkshake

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leading 1

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There is a kernel!

kernel = nukleus (Lat)

Does this mean, there was a nuklear or even a nukelar device in that milkshake?

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lets rowreduce this. just kidding...

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Row reduced echelon form

  • 3. Every row above leading row leads to the left
  • 1. Every first nonzero element in a row is 1
  • 2. Leading columns otherwise only contain 0’s

“Leaders like to be first, do not like other leaders in the same column and like leaders above them to be to their left. “

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1 1 1 4 2

Row reduced echelon form?

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1 1 4 2 6 2 4 2 3

Row reduced echelon form?

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1 1 3 4

Row reduced echelon form?

5 1 1 1 5

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Inverting a matrix

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Second blackboard problem

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3 4 3 1 1 1 10

invert the following matrix

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3 4 3 1 1 1 10

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1 1 1 3 4 3 1 1 1 10

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1 1 1 3 4 3 1 1 1 10 1

  • 3

10 2

  • 1

1

  • 3

1 1 1

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Which 2x2 matrices are their own inverse?

1 1 1

  • 1
  • 1 0
  • 1
  • 1 0

1

4 examples: are there more?

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There are more!

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Two hints: think geometrically change basis

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  • II. Linear transformations
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T(x+y) = T(x) + T(y) T(r x) = r T(x) T(0)=0

T plays well with 0, addition and scalar multiplication:

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How do we express T as a matrix?

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Key Fact: The columns of A are the images of the basis vectors.

v v v ...

1 2 m

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Geometry

v v v ...

1 2 m

Algebra

ei v i

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1 1 2 1

Example

What does the following transformation do?

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Examples of transformations

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rotations

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rotation-dilation

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shears

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projections

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Pikaboo, where has Oliver gone?

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reflections

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dilations

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Third blackboard problem

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Find the matrix of the transformation in R4 which reflects at the xz plane then reflects at the yz plane.

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Quiz coming up!

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6 -8 8 6 1

100

What is the length of

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The answer is ....

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10

100

1 gogool, term coined by Milton Sirotta (1929-1980), nephew of Edward Kasner (1878-1955) 1 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000

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  • III. Basis
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Linear independence

a v + a v + .... + a v = 0

1 1 2 2 n n

then, a = a = ... = a = 0

1 2 n

if

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Spanning

every v in V can be written as a v + a v + .... + a v = v

1 1 2 2 n n

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Basis

linear independent and span

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Standard basis

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Standard basis

1 1 1 1

e1 e2 e3 e4

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How do we check to have a basis?

v

1

v2 v3 v 4

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Fourth blackboard problem

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Is this a basis?

1 1 1 1 1 2 1 1 1 1 3 1 1 1 1 4

v

1

v2 v3 v 4

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  • IV. Image and Kernel
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Rene Magritte, The son of man: 1963

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Oliver Knill Son of a peach, 2007

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Oliver Knill Son of a peach, 2007

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ker(A) im(A)

im(A) = { Ax | x in V } ker(A) = { x | Ax =0 }

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How do we compute a basis for the image and kernel?

row reduce!

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Dimension formula

dim(im(A)) + dim(ker(A)) = m

fundamental theorem of linear algebra? rank nulletly theorem rank nullety

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Computing the image:

The basis is the set of pivot columns.

Computing the kernel:

The basis is obtained by solving the linear system in row reduced echelon form and taking free variables.

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Fifth blackboard problem

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Image and kernel of X matrix

1 1 1 1 1 1 1 1 1

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XXX (2003) , movie won Taurus award for this stunt. 18 cameras filmed in Auburn CA at 730 feet bridge.

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  • IV. Coordinates
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[v] = S v

  • 1

B

v coordinates in standard basis

[v] coordinates in basis B

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B = S A S

  • 1
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Sixth blackboard problem

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Problem

1 1 1 1 1 1 1 1 1 1

{ }

, , ,

B=

What are the B coordinates of v =

1 2 3 4

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1 1 1 1 1 1 1 1 1 1

S=

1

  • 1

1

  • 1

1

  • 1

1

S=

  • 1
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1

  • 1

1

  • 1

1

  • 1

1

Sv=

  • 1

1 2 3 4

=

  • 1
  • 1
  • 1

4

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  • V. Linear Spaces
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R

n

R

2

R

1

R

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From vectors to functions

4 1 3 4 2

10%

30%

40%

20%

1 2 3 4 1 2 3 4

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From vectors to functions

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

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polynomials

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Basis

P

n

has basis {1,x ,x ,.... x }

n 1 2

and so dim(P ) = n+1

n

Dimension: how many parameters do we need to describe a general object in the space?

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What is the dimension of

P

5

X ={f in | f(0)=0 }

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soltions to differential solutions to differential equations

X={ f in C (R) | f’’(x)=-f(x)}

Example: 8

Solutions to linear differential equations are linear spaces.

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matrices

+

1 2 4 6 1

  • 1

2 1 3 3 8

=

1 2

  • 1

2

  • 3

= -3 -6

3 6

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periodic functions

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Seventh blackboard problem

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Find a basis and dimension of

P

3

X ={f in | f(0)=1 }

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P

3

X ={f in | f(0)=1 }

Trap!

This is not a linear space!

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Find a basis and dimension of

P

3

X ={f in | f(0)=0 }

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The end

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