# Table of Contents

### Chapter 1  Vector Spaces

 1.1 Introduction 1.2 Vector Spaces 1.3 Subspaces 1.4 Linear Combinations and Systems of Linear Equations 1.5 Linear Dependence and Linear Independence 1.6 Bases and Dimension 1.7* Maximal Linearly Independent Subsets Index of Definitions

### Chapter 2  Linear Transformations and Matrices

 2.1 Linear Transformations, Null Spaces, and Ranges 2.2 The Matrix Representation of a Linear Transformation 2.3 Composition of Linear Transformations and Matrix Multiplication 2.4 Invertibility and Isomorphisms 2.5 The Change of Coordinate Matrix 2.6* Dual Spaces 2.7* Homogeneous Linear Differential Equations with Constant Coefficients Index of Definitions

### Chapter 3  Elementary Matrix Operations and Systems of Linear Equations

 3.1 Elementary Matrix Operations and Elementary Matrices 3.2 The Rank of a Matrix and Matrix Inverses 3.3 Systems of Linear Equations---Theoretical Aspects 3.4 Systems of Linear Equations---Computational Aspects Index of Definitions

### Chapter 4  Determinants

 4.1 Determinants of Order 2 4.2 Determinants of order n 4.3 Properties of Determinants 4.4 Summary---Important Facts about Determinants 4.5* A Characterization of the Determinant Index of Definitions

### Chapter 5  Diagonalization

 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalizability 5.3* Matrix Limits and Markov Chains 5.4 Invariant Subspaces and the Cayley-Hamilton Theorem Index of Definitions

### Chapter 6  Inner Product Spaces

 6.1 Inner Products and Norms 6.2 The Gram-Schmidt Orthogonalization Process and Orthogonal Complements 6.3 The Adjoint of a Linear Operator 6.4 Normal and Self-Adjoint Operators 6.5 Unitary and Orthogonal Operators and Their Matrices 6.6 Orthogonal Projections and the Spectral Theorem 6.7* The Singular Value Decomposition and the Pseudoinverse 6.8* Bilinear and Quadratic Forms 6.9* Einstein's Special Theory of Relativity 6.10* Conditioning and the Rayleigh Quotient 6.11* The Geometry of Orthogonal Operators Index of Definitions

### Chapter 7  Canonical Forms

 7.1 Jordan Canonical Form I 7.2 Jordan Canonical Form II 7.3 The Minimal Polynomial 7.4* Rational Canonical Form Index of Definitions

### Appendices

 A Sets B Functions C Fields D Complex Numbers E Polynomials

### Index

* Sections denoted by an asterisk are optional.

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