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.