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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 | |