Knowledge of linear algebra, which occupies a central place in modern mathematics, is essential for anyone studying such subjects as Galois theory, function spaces, homology, cohomology—or any other area of mathematics. Written for advanced students, Steven Weintraub’s A Guide to Advanced Linear Algebra is about vector spaces and linear transformations. Taking a theoretical approach to his topics, Weintraub offers proofs of all results. Except for briefly mentioning Hilbert matrices, the author does not treat computational issues.
1. Vector spaces and linear transformations
4. The structure of linear transformations I
5. The structure of linear transformations II
6. Bilinear sesquilinear, and quadratic forms
7. Real and complex product spaces
8. Matrix groups as Lie groups
B. Modules over principal ideal domains
We regard linear algebra as part of algebra, and that guides our approach. But we have followed a middle ground. One of the principal goals of this book is to derive canonical forms for linear transformations on finite dimensional vector spaces, i.e., rational and Jordan canonical forms. The quickest and perhaps most enlightening approach is to derive them as corollaries of the basic structure theorems for modules over a principal ideal domain (PID). Doing so would require a good deal of background, which would limit the utility of this book. Thus our main line of approach does not use these, though we indicate this approach in an appendix. Instead we adopt a more direct argument.
Steven H. Weintraub (Lehigh University) has written nine books and authored 50 papers. He has served on the executive committee of the Eastern Pennsylvania-Delaware section of the MAA.
The word “advanced” in the title of this book can be interpreted in two ways. On the one hand, the book discusses topics in linear algebra (canonical forms, bilinear and sesquilinear forms, matrix Lie groups, etc.) that are rarely taught in an introductory course. On the other hand, even when discussing topics that are usually considered elementary, the book does so in a theoretical, sophisticated way, generally eschewing the kind of routine calculations that are so common in introductory linear algebra books in favor of carefully defined terms and precise statements (and proofs) of theorems, often presented at a fairly high level of generality. Continued...