Contents
Preface iii
Matrices and Systems of Equations 1
1.1 Systems of Linear Equations 1
1.2 Row EchelonForm 11
1.3 Matrix Arithmetic 27
1.4 Matrix Algebra 44
1.5 Elementary Matrices 58
1.6 Partitioned Matrices 68
MATLAB R Exercises 77
ChapterTestA 81
ChapterTestB 82
2 Determinants 84
2.1 The Determinant of a Matrix 84
2.2 Properties of Determinants 91
2.3. AdditionalTopics and Applications 98
MATLAB Exercises 106
ChapterTestA 108
ChapterTestB 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
ChapterTestA 164
ChapterTestB 164
4 Linear Transformations 166
4.1 Definition and Examples 166
4.2 Matrix Representationsof LinearTransformations 175
4.3 Similarity 189
MATLAB Exercises 195
ChapterTestA 196
ChapterTestB 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
ChapterTestA 279
ChapterTestB 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 SingularValue Decomposition 337
6.6 QuadraticForms 351
6.7 Positive Definite Matrices 364
6.8 Nonnegative Matrices 372
MATLAB Exercises 378
ChapterTestA 384
ChapterTestB 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
8 Iterative Methods Web.
9 Canonical Forms Web.
Appendix: MATLAB 456
Bibliography 468
Answers to Selected Exercises 471
Index 485
. Optional sections. These sections are not prerequisites for any other sections of the book.
. Web: The supplemental Chapters 8 and 9 can be downloaded from the internet. See the section of the Preface on supplemental Web materials.
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