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

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. Example: A x =b, means b is the dot k product of the k’th row with x. 4

mxp nxm . nxp matrix = multiplication 5

Problem: Can we multiply a 4 x 5 matrix A with a 5x4 matrix B? in other words: Is A B defined? 6

Matrix algebra: except for two things With nxn matrices in general: A,B,C,D,... one can work as with numbers A B = B A A+B = B+A a (A+B)=aA + aB -1 A might not exist even A(B+C) = AB + AC for nonzero A A (B C) = (A B) C -1 -1 -1 (A B) = B A etc 7

A,B, arbitrary True or False? nxn matrices 2 2 (A + B) ( A - B) = A - B 2 3 4 -1 (1+A+A + A ) = (1-A ) (1-A) assuming (1-A) is invertible 1=I n 8

Row reduction 9

Gauss-Jordan elimination Subtract Scale Swap row a two from row rows other row S S S 10

First blackboard problem 11

Row reduce the X matrix 8 0 0 0 8 0 8 0 8 0 0 0 8 0 0 0 8 0 8 0 8 0 0 0 8 12

Remember the Boston milkshake scare? 13

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Gotscha! 15

You can no more escape 16

Executed with water gun 17

Warriors with their prey 18

The milkshake as a matrix 19

Lets row reduce it! 20

The rref death of a milkshake 21

leading 1 22

There is a kernel! kernel = nukleus (Lat) Does this mean, there was a nuklear or even a nukelar device in that milkshake? 23

lets rowreduce this. just kidding... 24

Row reduced echelon form 1. Every first nonzero element in a row is 1 2. Leading columns otherwise only contain 0’s 3. Every row above leading row leads to the left “Leaders like to be first, do not like other leaders in the same column and like leaders above them to be to their left. “ 25

Row reduced echelon form? 1 4 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 26

Row reduced echelon form? 1 4 2 6 2 0 1 2 3 4 0 0 0 0 0 0 0 0 0 0 27

Row reduced echelon form? 5 1 0 0 0 1 0 0 1 0 0 3 0 0 0 1 0 4 0 0 0 0 1 5 0 0 0 0 0 0 28

Inverting a matrix 29

Second blackboard problem 30

invert the following matrix 1 0 3 1 1 4 3 0 10 31

1 0 3 1 1 4 3 0 10 32

0 1 0 3 1 0 0 0 1 1 4 1 1 3 0 0 0 10 32

0 1 0 3 1 0 0 0 1 1 4 1 1 3 0 0 0 10 -3 1 10 0 0 0 -1 2 1 0 1 0 1 -3 0 0 0 1 32

Which 2x2 matrices are their own inverse? 1 0 -1 0 4 examples: 0 -1 0 1 1 0 -1 0 are there 0 1 0 -1 more? 33

There are more! 34

Two hints: think geometrically change basis 35

II. Linear transformations 36

T plays well with 0, addition and scalar multiplication: T(0)=0 T(x+y) = T(x) + T(y) T(r x) = r T(x) 37

How do we express T as a matrix? 38

Key Fact: The columns of A are the images of the basis vectors. v v ... v 1 2 m 39

Geometry Algebra v i v v ... v 1 2 m e i 40

Example What does the following transformation do? 1 0 0 0 1 2 0 0 1 41

Examples of transformations 42

rotations 43

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rotation-dilation 45

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shears 47

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projections 49

Pikaboo, where has Oliver gone? 50

reflections 51

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dilations 53

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Third blackboard problem 55

Find the matrix of the transformation in R 4 which reflects at the xz plane then reflects at the yz plane. 56

Quiz coming up! 57

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What is the length of 100 6 -8 1 8 6 0 59

The answer is .... 60

1 0000000000 0000000000 0000000000 0000000000 100 0000000000 10 0000000000 0000000000 0000000000 0000000000 0000000000 1 gogool, term coined by Milton Sirotta (1929-1980), nephew of Edward Kasner (1878-1955) 61

III. Basis 62

Linear independence if a v + a v + .... + a v = 0 1 1 2 2 n n then, a = a = ... = a = 0 1 2 n 63

Spanning every v in V can be written as a v + a v + .... + a v = v 1 1 2 2 n n 64

Basis linear independent and span 65

Standard basis 66

Standard basis 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 e 3 e 4 e 2 e 1 67

How do we check to have a basis? v v 2 v 3 v 4 1 68

Fourth blackboard problem 69

Is this a basis? 1 1 1 1 1 1 1 2 1 1 3 1 1 1 1 4 v v 2 v 3 v 4 1 70

IV. Image and Kernel 71

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Rene Magritte, The son of man: 1963 73

Oliver Knill Son of a peach, 2007 74

Oliver Knill Son of a peach, 2007 74

im(A) = { Ax | x in V } ker(A) im(A) ker(A) = { x | Ax =0 } 75

How do we compute a basis for the image and kernel? row reduce! 76

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Dimension formula rank nullety dim(im(A)) + dim(ker(A)) = m fundamental theorem of linear algebra? rank nulletly theorem 79

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. 80

Fifth blackboard problem 81

Image and kernel of X matrix 1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1 82

XXX (2003) , movie won Taurus award for this stunt. 18 cameras filmed in Auburn CA at 730 feet bridge. 83

IV. Coordinates 84

v coordinates in standard basis -1 [v] = S v B [v] coordinates in basis B 85

-1 B = S A S 86

Sixth blackboard problem 87

Problem { } , , , 1 1 1 1 B= 0 1 1 1 0 0 1 1 0 0 0 1 1 2 What are the B 3 coordinates of v = 4 88

1 1 1 1 S= 0 1 1 1 0 0 1 1 0 0 0 1 1 -1 0 0 -1 S= 0 1 -1 0 0 0 1 -1 0 0 0 1 89

1 -1 0 0 1 -1 Sv= 0 1 -1 0 2 0 0 1 -1 3 0 0 0 1 4 -1 = -1 -1 4 90

V. Linear Spaces 91

n R 0 R 2 R 1 R 92

From vectors 10% 20% to 30% 1 4 functions 3 40% 4 4 0 3 2 0 2 1 1 2 0 3 4 93

From 4 3 vectors 2 to 1 1 functions 2 0 3 4 4 3 0 2 0 1 1 0 2 3 4 94

polynomials 95