文档介绍：Contents Preface .......................ix 1. Linear Equations ..............1 Introduction . . . ............... 1 Gaussian Elimination and Matrices ........ 3 Gauss–Jordan Method ..............15 Two-Point BoundaryValue Problems .......18 Making Gaussian Elimination Work ........21 Ill-Conditioned Systems .............33 2. Rectangular Systems and Echelon Forms . . . 41 Row Echelon Form and Rank ...........41 Reduced Row Echelon Form ...........47 Consistencyof Linear Systems ..........53 Homogeneous Systems ..............57 Nonhomogeneous Systems ............64 Electrical Circuits . ...............73 3. Matrix Algebra .............. 79 From Ancient China to Arthur Cayley .......79 Addition and Transposition ...........81 Linearity ....................89 WhyDo It This Way ..............93 Matrix Multiplication ..............95 Properties of Matrix Multiplication ....... 105 Matrix Inversion . .............. 115 Inverses of Sums and Sensitivity ........ 124 ElementaryMatrices and Equivalence ...... 131 The LU Factorization ............. 141 4. Vector Spaces ...............159 Spaces and Subspaces ............. 159 Four Fundamental Subspaces .......... 169 Linear Independence ............. 181 Basis and Dimension ............. 194 vi Contents More about Rank . .............. 210 Classical Least Squares ............ 223 Linear Transformations ............ 238 Change of Basis and Similarity ......... 251 Invariant Subspaces .............. 259 5. Norms, Inner Products, and Orthogonality . . 269 Vector Norms . . .............. 269 Matrix Norms . . .............. 279 Inner-Product Spaces ............. 286 Orthogonal Vectors .............. 294 Gram–Schmidt Procedure ........... 307 Unitaryand Orthogonal Matrices ........ 320 Orthogonal Re