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| Preface to the Second Edition | |
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| Preface to the First Edition | |
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| The Algebra of Matrices | |
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| Matrices: Definitions | |
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| Addition and Scalar Multiplication of Matrices | |
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| Matrix Multiplication | |
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| Square Matrices, Inverses, and Zero Divisors | |
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| Transposes, Partitioning of Matrices, and Direct Sums | |
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| Linear Equations | |
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| Equivalent Systems of Equations | |
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| Row Operations on Matrices | |
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| Row Echelon Form | |
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| Homogeneous Systems of Equations | |
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| The Unrestricted Case: A Consistency Condition | |
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| The Unrestricted Case: A General Solution | |
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| Inverses of Nonsingular Matrices | |
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| Vector Spaces | |
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| Vectors and Vector Spaces | |
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| Subspaces and Linear Combinations | |
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| Linear Dependence and Linear Independence | |
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| Bases | |
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| Bases and Representations | |
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| Row Spaces of Matrices | |
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| Column Equivalence | |
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| Row-Column Equivalence | |
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| Equivalence Relations and Canonical Forms of Matrices | |
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| Determinants | |
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| Introduction as a Volume Function | |
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| Permutations and Permutation Matrices | |
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| Uniqueness and Existence of the Determinant Function | |
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| Practical Evaluation and Transposes of Determinants | |
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| Cofactors, Minors, and Adjoints | |
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| Determinants and Ranks | |
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| Linear Transformations | |
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| Definitions | |
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| Representation of Linear Transformations | |
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| Representations Under Change of Bases | |
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| Eigenvalues and Eigenvectors | |
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| Introduction | |
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| Relation Between Eigenvalues and Minors | |
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| Similarity | |
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| Algebraic and Geometric Multiplicites | |
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| Jordan Canonical Form | |
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| Functions of Matrices | |
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| Application: Markov Chains | |
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| Inner Produce Spaces | |
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| Inner Products | |
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| Representation of Inner Products | |
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| Orthogonal Bases | |
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| Unitary Equivalence and Hermitian Matrices | |
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| Congruence and Conjunctive Equivalence | |
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| Central Conics and Quadrics | |
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| The Natural Inverse | |
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| Normal Matrices | |
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| Applications to Differential Equations | |
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| Introduction | |
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| Homogeneous Differential Equations | |
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| Linear Differrential Equations: The Unrestricted Case | |
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| Linear Operators: The Global Vie? | |
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| Answers | |
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| Symbols | |
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| Index | |