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Preface | p. xi |
To the Student | p. xvii |
Matrices, Vectors, and Systems of Linear Equations | p. 3 |
Matrices and Vectors | p. 3 |
Linear Combinations, Matrix-Vector Products, and Special Matrices | p. 13 |
Systems of Linear Equations | p. 27 |
Gaussian Elimination | p. 41 |
Applications of Systems of Linear Equations | p. 56 |
The Span of a Set of Vectors | p. 66 |
Linear Dependence and Linear Independence | p. 75 |
Review Exercises | p. 87 |
MATLAB Exercises | p. 90 |
Matrices and Linear Transformations | p. 95 |
Matrix Multiplication | p. 95 |
Applications of Matrix Multiplication | p. 106 |
Invertibility and Elementary Matrices | p. 122 |
The Inverse of a Matrix | p. 135 |
Partitioned Matrices and Block Multiplication | p. 146 |
The LU Decomposition of a Matrix | p. 152 |
Linear Transformations and Matrices | p. 167 |
Composition and Invertibility of Linear Transformations | p. 180 |
Review Exercises | p. 192 |
MATLAB Exercises | p. 195 |
Determinants | p. 199 |
Cofactor Expansion | p. 199 |
Properties of Determinants | p. 211 |
Review Exercises | p. 222 |
MATLAB Exercises | p. 224 |
Subspaces and their Properties | p. 227 |
Subspaces | p. 227 |
Basis and Dimension | p. 241 |
The Dimension of Subspaces Associated with a Matrix | p. 254 |
Coordinate Systems | p. 263 |
Matrix Representations of Linear Operators | p. 275 |
Review Exercises | p. 286 |
MATLAB Exercises | p. 288 |
Eigenvalues, Eigenvectors, and Diagonalization | p. 293 |
Eigenvalues and Eigenvectors | p. 293 |
The Characteristic Polynomial | p. 302 |
Diagonalization of Matrices | p. 313 |
Diagonalization of Linear Operators | p. 325 |
Applications of Eigenvalues | p. 334 |
Review Exercises | p. 355 |
MATLAB Exercises | p. 356 |
Orthogonality | p. 361 |
The Geometry of Vectors | p. 361 |
Orthogonal Vectors | p. 374 |
Orthogonal Projections | p. 388 |
Least-Squares Approximation and Orthogonal Projection Matrices | p. 402 |
Orthogonal Matrices and Operators | p. 411 |
Symmetric Matrices | p. 425 |
Singular Value Decomposition | p. 438 |
Principal Component Analysis | p. 456 |
Rotations of R[superscript 3] and Computer Graphics | p. 465 |
Review Exercises | p. 481 |
MATLAB Exercises | p. 483 |
Vector Spaces | p. 489 |
Vector Spaces and Their Subspaces | p. 489 |
Linear Transformations | p. 501 |
Basis and Dimension | p. 511 |
Matrix Representations of Linear Operators | p. 521 |
Inner Product Spaces | p. 532 |
Review Exercises | p. 547 |
MATLAB Exercises | p. 548 |
Appendices | p. 551 |
Sets | p. 551 |
Functions | p. 553 |
Complex Numbers | p. 556 |
MATLAB | p. 561 |
The Uniqueness of the Reduced Row Echelon Form | p. 575 |
Bibliography | p. 579 |
Answers to Selected Exercises | p. 581 |
Index | p. 621 |
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