Advanced linear algebra

"What is linear algebra, and how is it used? Upon examining almost any introductory text on linear algebra, we find a standard list of topics that seems to define the subject. On one hand, one part of linear algebra consists of computational techniques for solving linear equations, multiplying and inverting matrices, calculating and interpreting determinants, finding eigenvalues and eigenvectors, and so on. On the other hand, there is a theoretical side to linear algebra involving abstract vector spaces, subspaces, linear independence, spanning sets, bases, dimension, and linear transformations. But there is much more to linear algebra than just vector spaces, matrices, and linear equations! The goal of this book is to explore a variety of advanced topics in linear algebra, which highlight the rich interconnections linking this subject to geometry, algebra, analysis, combinatorics, numerical computation, and many other areas of mathematics. The book consists of twenty chapters, grouped into six main subject areas (algebraic structures, matrices, structured matrices, geometric aspects of linear algebra, modules, and multilinear algebra). Some chapters approach introductory material from a more sophisticated or abstract viewpoint; other chapters provide elementary expositions of more theoretical concepts; yet other chapters offer unusual perspectives or novel treatments of standard results. Unlike some advanced mathematical texts, this book has been carefully designed to minimize the dependence of each chapter on material found in earlier chapters. Each chapter has been conceived as a "mathematical vignette" devoted to the development of one specific topic"--