Page 2 - Matrix Analysis & Applied Linear Algebra

Contents

Preface ....................... ix

1. Linear Equations .............. 1
1.1 Introduction . . . ............... 1
1.2 Gaussian Elimination and Matrices ........ 3
1.3 Gauss–Jordan Method .............. 15
1.4 Two-Point BoundaryValue Problems ....... 18
1.5 Making Gaussian Elimination Work ........ 21
1.6 Ill-Conditioned Systems ............. 33

2. Rectangular Systems and Echelon Forms . . . 41
2.1 Row Echelon Form and Rank ........... 41
2.2 Reduced Row Echelon Form ........... 47
2.3 Consistencyof Linear Systems .......... 53
2.4 Homogeneous Systems .............. 57
2.5 Nonhomogeneous Systems ............ 64
2.6 Electrical Circuits . ............... 73

3. Matrix Algebra .............. 79
3.1 From Ancient China to Arthur Cayley ....... 79
3.2 Addition and Transposition ........... 81
3.3 Linearity .................... 89
3.4 WhyDo It This Way .............. 93
3.5 Matrix Multiplication .............. 95
3.6 Properties of Matrix Multiplication ....... 105
3.7 Matrix Inversion . .............. 115
3.8 Inverses of Sums and Sensitivity ........ 124
3.9 ElementaryMatrices and Equivalence ...... 131
3.10 The LU Factorization ............. 141
4. Vector Spaces ............... 159

4.1 Spaces and Subspaces ............. 159
4.2 Four Fundamental Subspaces .......... 169
4.3 Linear Independence ............. 181
4.4 Basis and Dimension ............. 194