Linear Algebra Math 308 - University of Washington

Linear Algebra Math 308

S. Paul Smith

Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195

E-mail address: smith@math.washington.edu

Contents

Chapter 0. Introduction

1

1. What it's all about

1

2. Some practical applications

2

3. The importance of asking questions

2

Chapter 1. Linear Geometry

3

1. Linearity

3

2. Lines in R2

10

3. Points, lines, and planes in R3

13

4. Higher dimensions

15

5. Parametric descriptions of lines, planes, etc.

15

Chapter 2. Matrices

17

1. What is a matrix?

17

2. A warning about our use of the word vector

18

3. Row and column vectors = points with coordinates

18

4. Matrix arithmetic: addition and subtraction

20

5. The zero matrix and the negative of a matrix

20

6. Matrix arithmetic: multiplication

21

7. Pitfalls and warnings

25

8. Transpose

25

9. Some special matrices

26

10. Solving an equation involving an upper triangular matrix

27

11. Some special products

28

Chapter 3. Matrices and motions in R2 and R3

31

1. Linear transformations

31

2. Rotations in the plane

31

3. Projections in the plane

31

4. Contraction and dilation

31

5. Reflections in the plane

31

6. Reflections in R3

31

7. Projections from R3 to a plane

31

Chapter 4. Systems of Linear Equations

33

1. Systems of equations

33

2. A single linear equation

36

iii

iv

CONTENTS

3. Systems of linear equations

37

4. A system of linear equations is a single matrix equation

38

5. Specific examples

38

6. The number of solutions

41

7. A geometric view on the number of solutions

41

8. Homogeneous systems

42

Chapter 5. Row operations and row equivalence

45

1. Equivalent systems of equations

45

2. Echelon Form

45

3. An example

46

4. You already did this in high school

47

5. The rank of a matrix

48

6. Inconsistent systems

49

7. Consistent systems

49

8. Parametric equations for lines and planes

51

9. The importance of rank

52

10. The word "solution"

53

11. Elementary matrices

54

Chapter 6. The Vector space Rn

57

1. Arithmetic in Rn

57

2. The standard basis for Rn

58

3. Linear combinations and linear span

58

4. Some infinite dimensional vector spaces

60

Chapter 7. Subspaces

61

1. The definition and examples

61

2. The row and column spaces of a matrix

64

3. Lines, planes, and translations of subspaces

65

4. Linguistic difficulties: algebra vs. geometry

67

Chapter 8. Linear dependence and independence

69

1. The definition

69

2. Criteria for linear (in)dependence

69

3. Linear (in)dependence and systems of equations

73

Chapter 9. Non-singular matrices, invertible matrices, and inverses 75

1. Singular and non-singular matrices

75

2. Inverses

76

3. Elementary matrices are invertible

78

4. Making use of inverses

78

5. The inverse of a 2 ? 2 matrix

79

6. If A is non-singular how do we find A-1?

81

Chapter 10. Bases, coordinates, and dimension

83

1. Definitions

83

CONTENTS

v

2. A subspace of dimension d is just like Rd

83

3. All bases for W have the same number of elements

84

4. Every subspace has a basis

84

5. Properties of bases and spanning sets

85

6. How to find a basis for a subspace

86

7. How to find a basis for the range of a matrix

86

8. Rank + Nullity

86

9. How to compute the null space and range of a matrix

90

Chapter 11. Linear transformations

91

1. A reminder on functions

91

2. First observations

92

3. Linear transformations and matrices

94

4. How to find the matrix representing a linear transformation

95

5. Invertible matrices and invertible linear transformations

96

6. How to find the formula for a linear transformation

96

7. Rotations in the plane

96

8. Reflections in R2

97

9. Invariant subspaces

98

10. The one-to-one and onto properties

98

11. Two-to-two

98

12. Gazing into the distance: differential operators as linear

transformations

99

Chapter 12. Determinants

103

1. The definition

103

2. Elementary row operations and determinants

106

3. The determinant and invertibility

108

4. Properties

108

5. Elementary column operations and determinants

109

Chapter 13. Eigenvalues

113

1. Definitions and first steps

113

2. Reflections in R2, revisited

114

3. The 2 ? 2 case

115

4. The equation A2 + I

116

5. The characteristic polynomial

117

6. How to find eigenvalues and eigenvectors

118

7. The utility of eigenvectors

122

Chapter 14. Complex vector spaces and complex eigenvalues

125

1. The complex numbers

125

2. The complex numbers

125

3. Linear algebra over C

129

4. The complex norm

129

5. An extended exercise

131

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