Designed for advanced undergraduate and beginning graduate students in linear or abstract algebra, Advanced Linear Algebra covers theoretical aspects of the subject, along with examples, computations, and proofs. It 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 begins with chapters introducing basic notation for vector spaces, permutations, polynomials, and other algebraic structures. The following chapters are designed to be mostly independent of each other so that readers with different interests can jump directly to the topic they want. This is an unusual organization compared to many abstract algebra textbooks, which require readers to follow the order of chapters.
Each chapter consists of a mathematical vignette devoted to the development of one specific topic. Some chapters look at introductory material from a sophisticated or abstract viewpoint, while others provide elementary expositions of more theoretical concepts. Several chapters offer unusual perspectives or novel treatments of standard results.
A wide array of topics is included, ranging from concrete matrix theory (basic matrix computations, determinants, normal matrices, canonical forms, matrix factorizations, and numerical algorithms) to more abstract linear algebra (modules, Hilbert spaces, dual vector spaces, bilinear forms, principal ideal domains, universal mapping properties, and multilinear algebra).
The book provides a bridge from elementary computational linear algebra to more advanced, abstract aspects of linear algebra needed in many areas of pure and applied mathematics.
1. Overview of Algebraic Systems. 2. Permutations. 3. Polynomials. 4. Basic Matrix Operations. 5. Determinants via Calculations. 6. Comparing Concrete Linear Algebra to Abstract Linear Algebra. 7. Hermitian, Positive Definite, Unitary, and Normal Matrices. 8. Jordan Canonical Forms. 9. Matrix Factorizations. 10. Iterative Algorithms in Numerical Linear Algebra. 11. Affine Geometry and Convexity. 12. Ruler and Compass Constructions. 13. Dual Vector Spaces. 14. Bilinear Forms. 15. Metric Spaces and Hilbert Spaces. 16. Finitely Generated Commutative Groups. 17. Introduction to Modules. 18. Principal Ideal Domains, Modules over PIDs, and Canonical Forms. 19. Introduction to Universal Mapping Properties. 20. Universal Mapping Problems in Multilinear Algebra
Nicholas A. Loehr received his Ph.D. in mathematics from the University of California at San Diego in 2003, studying algebraic combinatorics under the guidance of Professor Jeffrey Remmel. After spending two years at the University of Pennsylvania as an NSF postdoc, Dr. Loehr taught mathematics at the College of William and Mary, the United States Naval Academy, and Virginia Tech. Dr. Loehr has authored over sixty refereed journal articles and three textbooks on combinatorics, advanced linear algebra, and mathematical proofs. He teaches classes in these subjects and many others, including cryptography, vector calculus, modern algebra, real analysis, complex analysis, and number theory.
