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| Preface | |
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| Linear Equations | |
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| Introduction | |
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| Gaussian Elimination and Matrices | |
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| Gauss-Jordan Method | |
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| Two-Point Boundary Value Problems | |
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| Making Gaussian Elimination Work | |
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| Ill-Conditioned Systems | |
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| Rectangular Systems and Echelon Forms | |
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| Row Echelon Form and Rank | |
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| Reduced Row Echelon Form | |
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| Consistency of Linear Systems | |
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| Homogeneous Systems | |
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| Nonhomogeneous Systems | |
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| Electrical Circuits | |
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| Matrix Algebra | |
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| From Ancient China to Arthur Cayley | |
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| Addition and Transposition | |
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| Linearity | |
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| Why Do It This Way | |
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| Matrix Multiplication | |
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| Properties of Matrix Multiplication | |
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| Matrix Inversion | |
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| Inverses of Sums and Sensitivity | |
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| Elementary Matrices and Equivalence | |
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| The LU Factorization | |
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| Vector Spaces | |
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| Spaces and Subspaces | |
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| Four Fundamental Subspaces | |
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| Linear Independence | |
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| Basis and Dimension | |
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| More about Rank | |
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| Classical Least Squares | |
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| Linear Transformations | |
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| Change of Basis and Similarity | |
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| Invariant Subspaces | |
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| Norms, Inner Products, and Orthogonality | |
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| Vector Norms | |
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| Matrix Norms | |
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| Inner-Product Spaces | |
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| Orthogonal Vectors | |
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| Gram-Schmidt Procedure | |
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| Unitary and Orthogonal Matrices | |
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| Orthogonal Reduction | |
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| Discrete Fourier Transform | |
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| Complementary Subspaces | |
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| Range-Nullspace Decomposition | |
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| Orthogonal Decomposition | |
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| Singular Value Decomposition | |
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| Orthogonal Projection | |
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| Why Least Squares? | |
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| Angles between Subspaces | |
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| Determinants | |
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| Determinants | |
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| Additional Properties of Determinants | |
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| Eigenvalues and Eigenvectors | |
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| Elementary Properties of Eigensystems | |
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| Diagonalization by Similarity Transformations | |
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| Functions of Diagonalizable Matrices | |
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| Systems of Differential Equations | |
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| Normal Matrices | |
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| Positive Definite Matrices | |
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| Nilpotent Matrices and Jordan Structure | |
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| Jordan Form | |
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| Functions of Nondiagonalizable Matrices | |
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| Difference Equations, Limits, and Summability | |
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| Minimum Polynomials and Krylov Methods | |
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| Perron-Frobenius Theory | |
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| Introduction | |
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| Positive Matrices | |
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| Nonnegative Matrices | |
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| Stochastic Matrices and Markov Chains | |
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| Index | |