David Cherney, Tom Denton, Rohit Thomas and Andrew …

Linear Algebra

David Cherney, Tom Denton, Rohit Thomas and Andrew Waldron

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Edited by Katrina Glaeser and Travis Scrimshaw First Edition. Davis California, 2013.

This work is licensed under a Creative Commons Attribution-NonCommercialShareAlike 3.0 Unported License.

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Contents

1 What is Linear Algebra?

9

1.1 Organizing Information . . . . . . . . . . . . . . . . . . . . . . 9

1.2 What are Vectors? . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 What are Linear Functions? . . . . . . . . . . . . . . . . . . . 15

1.4 So, What is a Matrix? . . . . . . . . . . . . . . . . . . . . . . 20

1.4.1 Matrix Multiplication is Composition of Functions . . . 25

1.4.2 The Matrix Detour . . . . . . . . . . . . . . . . . . . . 26

1.5 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 Systems of Linear Equations

37

2.1 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . 37

2.1.1 Augmented Matrix Notation . . . . . . . . . . . . . . . 37

2.1.2 Equivalence and the Act of Solving . . . . . . . . . . . 40

2.1.3 Reduced Row Echelon Form . . . . . . . . . . . . . . . 40

2.1.4 Solution Sets and RREF . . . . . . . . . . . . . . . . . 45

2.2 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.3 Elementary Row Operations . . . . . . . . . . . . . . . . . . . 52

2.3.1 EROs and Matrices . . . . . . . . . . . . . . . . . . . . 52

2.3.2 Recording EROs in (M |I ) . . . . . . . . . . . . . . . . 54

2.3.3 The Three Elementary Matrices . . . . . . . . . . . . . 56

2.3.4 LU , LDU , and P LDU Factorizations . . . . . . . . . . 58

2.4 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 61

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2.5 Solution Sets for Systems of Linear Equations . . . . . . . . . 63 2.5.1 The Geometry of Solution Sets: Hyperplanes . . . . . . 64 2.5.2 Particular Solution + Homogeneous Solutions . . . . . 65 2.5.3 Solutions and Linearity . . . . . . . . . . . . . . . . . . 66

2.6 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 68

3 The Simplex Method

71

3.1 Pablo's Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.2 Graphical Solutions . . . . . . . . . . . . . . . . . . . . . . . . 73

3.3 Dantzig's Algorithm . . . . . . . . . . . . . . . . . . . . . . . 75

3.4 Pablo Meets Dantzig . . . . . . . . . . . . . . . . . . . . . . . 78

3.5 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 80

4 Vectors in Space, n-Vectors

83

4.1 Addition and Scalar Multiplication in Rn . . . . . . . . . . . . 84

4.2 Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3 Directions and Magnitudes . . . . . . . . . . . . . . . . . . . . 88

4.4 Vectors, Lists and Functions: RS . . . . . . . . . . . . . . . . 94

4.5 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 97

5 Vector Spaces

101

5.1 Examples of Vector Spaces . . . . . . . . . . . . . . . . . . . . 102

5.1.1 Non-Examples . . . . . . . . . . . . . . . . . . . . . . . 106

5.2 Other Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.3 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 109

6 Linear Transformations

111

6.1 The Consequence of Linearity . . . . . . . . . . . . . . . . . . 112

6.2 Linear Functions on Hyperplanes . . . . . . . . . . . . . . . . 114

6.3 Linear Differential Operators . . . . . . . . . . . . . . . . . . . 115

6.4 Bases (Take 1) . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.5 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 118

7 Matrices

121

7.1 Linear Transformations and Matrices . . . . . . . . . . . . . . 121

7.1.1 Basis Notation . . . . . . . . . . . . . . . . . . . . . . 121

7.1.2 From Linear Operators to Matrices . . . . . . . . . . . 127

7.2 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 129

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7.3 Properties of Matrices . . . . . . . . . . . . . . . . . . . . . . 133 7.3.1 Associativity and Non-Commutativity . . . . . . . . . 140 7.3.2 Block Matrices . . . . . . . . . . . . . . . . . . . . . . 142 7.3.3 The Algebra of Square Matrices . . . . . . . . . . . . 143 7.3.4 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7.4 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.5 Inverse Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.5.1 Three Properties of the Inverse . . . . . . . . . . . . . 150 7.5.2 Finding Inverses (Redux) . . . . . . . . . . . . . . . . . 151 7.5.3 Linear Systems and Inverses . . . . . . . . . . . . . . . 153 7.5.4 Homogeneous Systems . . . . . . . . . . . . . . . . . . 154 7.5.5 Bit Matrices . . . . . . . . . . . . . . . . . . . . . . . . 154 7.6 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.7 LU Redux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.7.1 Using LU Decomposition to Solve Linear Systems . . . 160 7.7.2 Finding an LU Decomposition. . . . . . . . . . . . . . 162 7.7.3 Block LDU Decomposition . . . . . . . . . . . . . . . . 165 7.8 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 166

8 Determinants

169

8.1 The Determinant Formula . . . . . . . . . . . . . . . . . . . . 169

8.1.1 Simple Examples . . . . . . . . . . . . . . . . . . . . . 169

8.1.2 Permutations . . . . . . . . . . . . . . . . . . . . . . . 170

8.2 Elementary Matrices and Determinants . . . . . . . . . . . . . 174

8.2.1 Row Swap . . . . . . . . . . . . . . . . . . . . . . . . . 175

8.2.2 Row Multiplication . . . . . . . . . . . . . . . . . . . . 176

8.2.3 Row Addition . . . . . . . . . . . . . . . . . . . . . . . 177

8.2.4 Determinant of Products . . . . . . . . . . . . . . . . . 179

8.3 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 182

8.4 Properties of the Determinant . . . . . . . . . . . . . . . . . . 186

8.4.1 Determinant of the Inverse . . . . . . . . . . . . . . . . 190

8.4.2 Adjoint of a Matrix . . . . . . . . . . . . . . . . . . . . 190

8.4.3 Application: Volume of a Parallelepiped . . . . . . . . 192

8.5 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 193

9 Subspaces and Spanning Sets

195

9.1 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9.2 Building Subspaces . . . . . . . . . . . . . . . . . . . . . . . . 197

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