* Sections denoted by an asterisk are optional.

• 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

• 2.1 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

• 3.1 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

• 4.1 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

• 5.1 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

• 6.1 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

• 7.1 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