Linear algebra I. The foundations for students of mathematics and physics. (Lineare Algebra I. Die Grundlagen für Studierende der Mathematik und Physik.) (German) Zbl 1357.15001

Waldmann, Stefan

Heidelberg: Springer Spektrum (ISBN 978-3-662-49913-9/pbk; 978-3-662-49914-6/ebook). xvii, 455 p. (2017).

This is the first of a two-volume text on linear algebra. The second deals mainly with multilinear algebra. Linear algebra textbooks cater for basically two types of students: those with only a rudimentary knowledge of mathematics who wish to apply matrix methods to various situations, and those with a solid mathematical foundation who wish to have a rigorous introduction to linear algebra. This book is definitely for the latter. But here, the choice is very large, so how does it differ from others? The major difference is that it takes the student on a wide tour of foundations. Thus, instead of assuming an acquaintance with \(\mathbb{R}\) and then briefly introducing \(\mathbb{C}\), the author starts with basic set theory and introduces semigroups, monoids, groups, rings, and fields, with examples. Moreover, the terms introduced are used throughout the text. There is also a section on introductory logic (e.g. quantifiers and induction). All this is of real benefit to those who wish to have a reference source for these concepts. On the other hand, it often encumbers the statements and proofs with not strictly necessary notation. Thus, this book is at the same time elementary and advanced – elementary in that the pace is gentle and everything is fully explained, advanced in that the setting is very mathematical. Chapters 1 and 2 comprise a summary of vector techniques in \(\mathbb{R}^3\). The remaining chapters are: 3. Groups, rings and fields, 4. Linear systems of equations and vector spaces, 5. Linear operators and matrices, 6. Determinants and eigenvalues, 7. Euclidean and unitary spaces, Appendix A: Basic concepts of logic, Appendix B: Sets and maps. Among the more advanced topics, there are dual spaces, minimal polynomial, Jordan canonical form, spectral theorem for normal operators, positive operators, polar decomposition, singular value decomposition, and operator norm. There are more than 200 exercises, most of them theoretical and introducing further facts.
With its 450 pages, this is a long book, packed with a great deal of information, and it will be appreciated by those who need a deeper insight. But it does not always provide a direct path to some of the key results because the way is paved with many interim results. The reader will, however, be amply rewarded by following those paths.

Reviewer: Rabe von Randow (Bonn)

MSC:

15-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra
00A05	Mathematics in general
00A06	Mathematics for nonmathematicians (engineering, social sciences, etc.)
00A35	Methodology of mathematics
15A23	Factorization of matrices
15A60	Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A03	Vector spaces, linear dependence, rank, lineability
15A06	Linear equations (linear algebraic aspects)
15A15	Determinants, permanents, traces, other special matrix functions
15A18	Eigenvalues, singular values, and eigenvectors
15A21	Canonical forms, reductions, classification
15B48	Positive matrices and their generalizations; cones of matrices
15B57	Hermitian, skew-Hermitian, and related matrices
65F20	Numerical solutions to overdetermined systems, pseudoinverses

Keywords:

linear algebra; introduction; textbook; introductory logic; groups; rings; fields; linear systems of equations; vector spaces; linear operators; matrices; determinants; eigenvalues; Euclidean space; unitary space; dual space; minimal polynomial; Jordan canonical form; spectral theorem; normal operators; positive operators; polar decomposition; singular value decomposition; operator norm