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



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



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



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



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



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



7 Canonical Forms



7.1 The Jordan Canonical Form I

7.2 The Jordan Canonical Form II

7.3 The Minimal Polynomial

7.4* The Rational Canonical Form

Index of Definitions



Appendices

A. Sets

B. Functions

C. Fields

D. Complex Numbers

E. Polynomials




Answers to Selected Exercises




Index