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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