Designed for a second theoretical course in linear algebra, this text explores a variety of advanced topics in linear algebra that highlight the rich interconnections of the subject to geometry, algebra, analysis, combinatorics, numerical computation, and many other areas of mathematics. The author covers algebraic structures, matrices, structured matrices, geometric aspects of linear algebra, modules, and multilinear algebra. He gives complete proofs of all results and supplements the general theory with many specific examples and concrete computations.

Background on Algebraic Structures. Overview of Algebraic Systems. Permutations. Polynomials. Matrices. Basic Matrix Operations. Determinants via Calculations. Concrete vs. Abstract Linear Algebra. Matrices with Special Structure. Hermitian, Positive Definite, Unitary, and Normal Matrices. Jordan Canonical Forms. Matrix Factorizations. Iterative Algorithms in Numerical Linear Algebra. The Interplay of Geometry and Linear Algebra. Affine Geometry and Convexity. Ruler and Compass Constructions. Dual Spaces and Bilinear Forms. Metric Spaces and Hilbert Spaces. Modules, Independence, and Classification Theorems. Finitely Generated Commutative Groups. Axiomatic Approach to Independence, Bases, and Dimension. Elements of Module Theory. Principal Ideal Domains, Modules over PID's, and Canonical Forms. Universal Mapping Properties and Multilinear Algebra. Introduction to Universal Mapping Properties. Universal Mapping Problems in Multilinear Algebra. Appendix: Basic Definitions. Sets. Functions. Relations. Partially Ordered Sets. Further Reading. Bibliography. Index.