1 Matrices and Systems of Equations 1
1.1 Systems of Linear Equations 1
1.2 Row Echelon Form 11
1.3 Matrix Arithmetic 27
1.4 Matrix Algebra 44
1.5 Elementary Matrices 58
1.6 Partitioned Matrices 68
MATLAB？ Exercises 77
Chapter Test A 81
Chapter Test B 82
2 Determinants 84
2.1 The Determinant of a Matrix 84
2.2 Properties of Determinants 91
2.3 Additional Topics and Applications 98
MATLAB Exercises 106
Chapter Test A 108
Chapter Test B 108
3 Vector Spaces 110
3.1 Definition and Examples 110
3.2 Subspaces 117
3.3 Linear Independence 127
3.4 Basis and Dimension 138
3.5 Change of Basis 144
3.6 Row Space and Column Space 154
MATLAB Exercises 162
Chapter Test A 164
Chapter Test B 164
4 Linear Transformations 166
4.1 Definition and Examples 166
4.2 Matrix Representations of Linear Transformations 175
4.3 Similarity 189
MATLAB Exercises 195
Chapter Test A 196
Chapter Test B 197
5 Orthogonality 198
5.1 The Scalar Product in Rn 199
5.2 Orthogonal Subspaces 214
5.3 Least Squares Problems 222
5.4 Inner Product Spaces 232
5.5 Orthonormal Sets 241
5.6 The Gram-Schmidt Orthogonalization Process 259
5.7 Orthogonal Polynomials 269
MATLAB Exercises 277
Chapter Test A 279
Chapter Test B 280
6 Eigenvalues 282
6.1 Eigenvalues and Eigenvectors 283
6.2 Systems of Linear Differential Equations 296
6.3 Diagonalization 307
6.4 Hermitian Matrices 324
6.5 The Singular Value Decomposition 337
6.6 Quadratic Forms 351
6.7 Positive Definite Matrices 364
6.8 Nonnegative Matrices 372
MATLAB Exercises 378
Chapter Test A 384
Chapter Test B 384
7 Numerical Linear Algebra 386
7.1 Floating-Point Numbers 387
7.2 Gaussian Elimination 391
7.3 Pivoting Strategies 398
7.4 Matrix Norms and Condition Numbers 403
7.5 Orthogonal Transformations 417
7.6 The Eigenvalue Problem 428
7.7 Least Squares Problems 437
MATLAB Exercises 448
Chapter Test A 454
Chapter Test B 454
Appendix： MATLAB 456
Bibliography 468
Answers to Selected Exercises 471
Index 485