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
