| Matrices and Gaussian Elimination | p. 1 |
| Introduction | p. 1 |
| The Geometry of Linear Equations | p. 3 |
| An Example of Gaussian Elimination | p. 11 |
| Matrix Notation and Matrix Multiplication | p. 19 |
| Triangular Factors and Row Exchanges | p. 32 |
| Inverses and Transposes | p. 45 |
| Special Matrices and Applications | p. 58 |
| Review Exercises: Chapter 1 | p. 65 |
| Vector Spaces | p. 69 |
| Vector Spaces and Subspaces | p. 69 |
| Solving Ax = 0 and Ax = b | p. 77 |
| Linear Independence, Basis, and Dimension | p. 92 |
| The Four Fundamental Subspaces | p. 102 |
| Graphs and Networks | p. 114 |
| Linear Transformations | p. 125 |
| Review Exercises: Chapter 2 | p. 137 |
| Orthogonality | p. 141 |
| Orthogonal Vectors and Subspaces | p. 141 |
| Cosines and Projections onto Lines | p. 152 |
| Projections and Least Squares | p. 160 |
| Orthogonal Bases and Gram-Schmidt | p. 174 |
| The Fast Fourier Transform | p. 188 |
| Review Exercises: Chapter 3 | p. 198 |
| Determinants | p. 201 |
| Introduction | p. 201 |
| Properties of the Determinant | p. 203 |
| Formulas for the Determinant | p. 210 |
| Applications of Determinants | p. 220 |
| Review Exercises: Chapter 4 | p. 230 |
| Eigenvalues and Eigenvectors | p. 233 |
| Introduction | p. 233 |
| Diagonalization of a Matrix | p. 245 |
| Difference Equations and Powers A[superscript k] | p. 254 |
| Differential Equations and e[superscript At] | p. 266 |
| Complex Matrices | p. 280 |
| Similarity Transformations | p. 293 |
| Review Exercises: Chapter 5 | p. 307 |
| Positive Definite Matrices | p. 311 |
| Minima, Maxima, and Saddle Points | p. 311 |
| Tests for Positive Definiteness | p. 318 |
| Singular Value Decomposition | p. 331 |
| Minimum Principles | p. 339 |
| The Finite Element Method | p. 346 |
| Computations with Matrices | p. 351 |
| Introduction | p. 351 |
| Matrix Norm and Condition Number | p. 352 |
| Computation of Eigenvalues | p. 359 |
| Iterative Methods for Ax = b | p. 367 |
| Linear Programming and Game Theory | p. 377 |
| Linear Inequalities | p. 377 |
| The Simplex Method | p. 382 |
| The Dual Problem | p. 392 |
| Network Models | p. 401 |
| Game Theory | p. 408 |
| Intersection, Sum, and Product of Spaces | p. 415 |
| The Jordan Form | p. 422 |
| Solutions to Selected Exercises | p. 428 |
| Matrix Factorizations | p. 474 |
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