A Guide to Advanced Linear Algebra

Linear algebra occupies a central place in pure mathematics. It plays an essential role in such widely differing fields as Galois theory, function spaces and homological algebra. In this context linear algebra is about vector spaces and linear transformations, not about matrices. The natural and perhaps most enlightening approach to the canonical forms for linear transformations on finite-dimensional vector spaces, one of the main goals of this book, is via the basic structure theorems for modules over a principal ideal domain, but the author has not gone that far. The infinite-dimensional case is, however, treated. Thus, the book is for advanced students of pure mathematics and not for those requiring a textbook on numerically applicable linear algebra. It is, moreover, written in a style to which a student of pure mathematics is fully accustomed.

Although the book is for advanced students, it begins with the basics, but it does not deal with matrix operations or the solution of systems of linear equations. The chapter headings are: (1) Vector spaces and linear transformations, (2) Coordinates, (3) Determinants, (4) and (5) The structure of a linear transformation I and II, (6) Bilinear, sesquilinear and quadratic forms, (7) Real and complex inner product spaces, (8) Matrix groups as Lie groups, Appendix A: Polynomials. Appendix B: Modules over principal ideal domains.

This book can be warmly recommended to any student of pure mathematics requiring a precise and concise treatment of all the important and “well-known” results of linear algebra. The student will be grateful for the direct route that it follows and for the occasional explanations of the “right” way to understand the material. –Rabe von Randow, Zentrallblatt