• * Sections denoted by an asterisk are optional.

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 CayleyHamilton Theorem
   • Index of Definitions
6. Inner Product Spaces
   • 6.1 Inner Products and Norms
   • 6.2 The GramSchmidt 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

• A. Sets
• B. Functions
• C. Fields
• D. Complex Numbers
• E. Polynomials

Answers to Selected Exercises

Index