Advanced linear algebra (Q2873811)

First, let us have a look at the contents. The book under review is divided into twenty chapters falling into six parts as follows: I~Background on algebraic structures: 1.~Overview of algebraic systems. 2.~Permutations. 3.~Polynomials. II~Matrices: 4.~Basic matrix operations. 5~Determinants via calculations. 6.~Concrete vs.\ abstract linear algebra. III~Matrices with special structure: 7.~Hermitian, positive definite, unitary, and normal matrices. 8.~Jordan canonical forms. 9.~Matrix factorisations. 10~Iterative algorithms in numerical linear algebra. IV~The interplay of geometry and linear algebra: 11.~Affine geometry and convexity. 12.~Ruler and compass constructions. 13.~Dual spaces and bilinear forms. 14.~Metric spaces and Hilbert spaces. V~Modules, independence, and classification theorems: 15.~Finitely generated commutative groups. 16.~Axiomatic approach to independence, bases, and dimension. 17.~Elements of module theory. 18.~Principal ideal domains, modules over PIDs, and canonical forms. VI~Universal mapping properties and multilinear algebra: 19.~Introduction to universal mapping properties. 20.~Universal mapping problems in multilinear algebra.NEWLINENEWLINEThe above-noted captions do not only sketch the topics covered by the book, but they indicate also the main idea behind: To offer a guided tour through a wealth of areas, starting at the very beginning with algebraic basics, going over to elementary linear algebra (in terms of coordinates), switching to the coordinate-free approach, and -- whenever a hilltop has been reached -- to take a detailed look at some neighbouring areas of mathematics, where linear algebra is indispensable.NEWLINENEWLINEThe selection of those ``neighbouring'' topics clearly reflects the author's preferences. Thus the book may not always present what one would expect according to the captions. For example, in Part IV nothing is said about projective geometry which, in the reviewer's humble opinion, lies at the crossover point of geometry and linear algebra.NEWLINENEWLINEThe exposition usually provides a twofold approach: On the one hand there are informal descriptions and explanations, like ``a matrix is a rectangular array'', and on the other hand there are precise formal definitions and complete proofs. The appealing text is rounded off by many accompanying specific examples and concrete computations. Illustrations, though, are rare. Every chapter closes with a detailed summary and many exercises with different levels of difficulty. Detailed suggestions for further reading are collected in an appendix at the end of the book.NEWLINENEWLINEAll things considered, it was a pleasure to read this carefully prepared book.