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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 | |