Linear algebra done right. 2nd ed. (English) Zbl 0886.15001

Axler, Sheldon

Undergraduate Texts in Mathematics. New York, NY: Springer. xii, 251 p. (1997).

[See the review of the first edition (1995; Zbl 0843.15002).]
This second edition of an almost determinant-free, none the less remarkably far-reaching and didactically masterly undergraduate text on linear algebra has undergone some substantial improvements. First of all, the sections on selfadjoint operators, normal operators, and the spectral theorem have been rewritten, methodically rearranged, and thus evidently simplified. Secondly, the section on orthogonal projections on inner-product spaces has been extended by taking up the application to minimization problems in geometry and analysis. Furthermore, several proofs have been simplified, and incidentally made more general and elegant (e.g., the proof of the trigonalizability of operators on finite-dimensional complex vector spaces, or the proof of the existence of a Jordan normal form for a nilpotent operator). Finally, apart from many other minor improvements and corrections throughout the entire text, several new examples and new exercises have been worked in. However, no mitigation has been granted to determinants. Altogether, with the present second edition of his text, the author has succeeded to make this an even better book.

Reviewer: W.Kleinert (Berlin)

MSC:

15-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra
15A03	Vector spaces, linear dependence, rank, lineability
15A04	Linear transformations, semilinear transformations
15A21	Canonical forms, reductions, classification
15A18	Eigenvalues, singular values, and eigenvectors
15A63	Quadratic and bilinear forms, inner products
15A15	Determinants, permanents, traces, other special matrix functions

Keywords:

vector spaces; linear maps; eigenvectors; matrices; canonical forms; inner-product spaces; orthogonal projections; normal operators; spectral theory; traces; determinants; textbook; exercises

Citations:

Zbl 0843.15002