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| Matrices and Gaussian Elimination | |
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| Introduction | |
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| The Geometry of Linear Equations | |
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| An Example of Gaussian Elimination | |
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| Matrix Notation and Matrix Multiplication | |
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| Triangular Factors and Row Exchanges | |
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| Inverses and Transposes | |
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| Special Matrices and Applications | |
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| Review Exercises: Chapter 1 | |
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| Vector Spaces | |
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| Vector Spaces and Subspaces | |
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| Solving Ax = 0 and Ax = b | |
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| Linear Independence, Basis, and Dimension | |
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| The Four Fundamental Subspaces | |
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| Graphs and Networks | |
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| Linear Transformations | |
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| Review Exercises: Chapter 2 | |
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| Orthogonality | |
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| Orthogonal Vectors and Subspaces | |
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| Cosines and Projections onto Lines | |
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| Projections and Least Squares | |
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| Orthogonal Bases and Gram-Schmidt | |
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| The Fast Fourier Transform | |
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| Review Exercises: Chapter 3 | |
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| Determinants | |
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| Introduction | |
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| Properties of the Determinant | |
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| Formulas for the Determinant | |
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| Applications of Determinants | |
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| Review Exercises: Chapter 4 | |
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| Eigenvalues and Eigenvectors | |
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| Introduction | |
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| Diagonalization of a Matrix | |
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| Difference Equations and Powers A[superscript k] | |
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| Differential Equations and e[superscript At] | |
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| Complex Matrices | |
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| Similarity Transformations | |
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| Review Exercises: Chapter 5 | |
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| Positive Definite Matrices | |
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| Minima, Maxima, and Saddle Points | |
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| Tests for Positive Definiteness | |
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| Singular Value Decomposition | |
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| Minimum Principles | |
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| The Finite Element Method | |
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| Computations with Matrices | |
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| Introduction | |
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| Matrix Norm and Condition Number | |
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| Computation of Eigenvalues | |
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| Iterative Methods for Ax = b | |
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| Linear Programming and Game Theory | |
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| Linear Inequalities | |
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| The Simplex Method | |
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| The Dual Problem | |
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| Network Models | |
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| Game Theory | |
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| Intersection, Sum, and Product of Spaces | |
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| The Jordan Form | |
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| Solutions to Selected Exercises | |
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| Matrix Factorizations | |