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Preface | |
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List of Applications | |
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Introduction to Linear Equations and Matrices | |
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Introduction to Linear Systems and Matrices | |
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Gaussian Elimination | |
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The Algebra of Matrices: Four Descriptions of the Product | |
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Inverses and Elementary Matrices | |
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Gaussian Elimination as a Matrix Factorization | |
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Transposes, Symmetry, and Band Matrices: An Application | |
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Numerical and Programming Considerations: Partial Pivoting, Overwriting Matrices, and Ill-Conditioned Systems | |
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Review Exercises | |
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Determinants | |
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The Determinant Function | |
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Properties of Determinants | |
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Finding det A Using Signed Elementary Products | |
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Cofactor Expansion: Cramer's Rule | |
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Applications | |
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Review Exercises | |
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Vector Spaces | |
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Vectors in 2- and 3-Spaces | |
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Euclidean n-Space | |
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General Vector Spaces | |
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Subspaces, Span, Null Spaces | |
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Linear Independence | |
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Basis and Dimension | |
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The Fundamental Subspaces of a Matrix | |
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Rank | |
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Coordinates and Change of Basis | |
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An Application: Error-Correcting Codes | |
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Review Exercises | |
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Cumulative Review Exercises | |
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Linear Transformations, Orthogonal Projections and Least Squares | |
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Matrices as Linear Transformation | |
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Relationships Involving Inner Products | |
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Least Squares and Orthogonal Projections | |
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Orthogonal Bases and the Gram-Schmidt Process | |
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Orthogonal Matrices, QR Decompositions, and Least Squares (Revisited) | |
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Encoding the QR Decompositions: A Geometric Approach | |
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General Matrices of Linear of Linear Transformations | |
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Similarity | |
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Review Exercises | |
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Cumulative Review Exercises | |
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Eigenvectors and Eigenvalues | |
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A Brief Introduction to Determinants | |
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Eigenvalues and Eigenvectors | |
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Diagonalization | |
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Symmetric Matrices | |
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An Application - Difference Equations: Fibonacci Sequences and Markov Processes | |
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An Application -Differential Equations | |
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An Application -- Quadratic Forms | |
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Solving the Eigenvalue Problem Numerically | |
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Review Exercises | |
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Cumulative Review Exercises | |
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Further Directions | |
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Function Spaces | |
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The Singular Value Decomposition -- Generalized Inverses, the General Least-Squares Problem, and an Approach to Ill-Conditioned Systems | |
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Iterative Method | |
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Matrix Norms | |
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General Vector Spaces and Linear Transformations Over an Arbitrary Field | |
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Review Exercises | |
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More on LU Decompositions | |
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Counting Operations and Gauss-Jordan Elimination | |
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Another Application | |
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Introduction to MATLAB and Projects | |
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Bibliography and Further Readings | |
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Index | |