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Linear Algebra presents a careful treatment of the principal topics of linear algebra. This acclaimed theorem-proof text emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate. Applications to such areas as differential equations, economics, geometry and physics appear throughout. It is especially suited to a second course in linear algebra that emphasizes abstract vector spaces, and can be used in a first course with a strong theoretical emphasis. Updates to the 5th Edition include revised proofs of some theorems, additional examples, and new exercises. Also new in this revision are online solutions for selected theoretical exercises, accessible by short URLs at point-of-use.

Table of Contents
* Sections denoted by an asterisk are optional.

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
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
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
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
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
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
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
Sets
Functions
Fields
Complex Numbers
Polynomials
Answers to Selected Exercises
Index