1 Basic Theory 1.1 Induction and Well—Ordering 1.2 Elementary Linear Algebra 1.3 Fields 1.4 Vector Spaces 1.5 Bases 1.6 Linear Maps 1.7 Linear Maps as Matrices 1.8 Dimension and lsomorphism 1.9 Matrix Representations Revisited 1.10 Subspaces 1.11 Linear Maps and Subspaces 1.12 Linear Independence 1.13 Row Reduction 1.14 Dual Spaces 1.15 Quotient Spaces 2 Linear Operators 2.1 Polynomials 2.2 Linear Differential Equations 2.3 Eigenvalues 2.4 The Minimal Polynomial 2.5 Diagonalizability 2.6 Cyclic Subspaces 2.7 The Frobenius Canonical Form 2.8 The Jordan Canonical Form 2.9 The Smith Normal Form 3 Inner Product Spaces 3.1 Examples of Inner Products 3.1.1 Real Inner Products 3.1.2 Complex Inner Products 3.1.3 A Digression on Quaternions 3.2 Inner Products 3.3 Orthonormal Bases 3.4 Orthogonal Complements and Projections 3.5 Adjoint Maps 3.6 Orthogonal Projections Revisited 3.7 Matrix Exponentials 4 Linear Operators on Inner Product Spaces 4.1 Self—Adjoint Maps 4.2 Polarization and Isometries 4.3 The Spectral Theorem 4.4 Normal Operators 4.5 Unitary Equivalence 4.6 Real Forms 4.7 Orthogonal Transformations 4.8 Triangulability 4.9 The Singular Value Decomposition 4.10 The Polar Decomposition 4.11 Quadratic Forms 5 Determinants
5.1 Geometric Approach 5.2 Algebraic Approach 5.3 How to Calculate Volumes 5.4 Existence of the Volume Form 5.5 Determinants of Linear Operators 5.6 Linear Equations 5.7 The Characteristic Polynomial 5.8 Differential Equations References Index
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