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Advanced linear and matrix algebra. (English) Zbl 1471.15001

Cham: Springer (ISBN 978-3-030-52814-0/hbk; 978-3-030-52817-1/pbk; 978-3-030-52815-7/ebook). xvi, 494 p. (2021).
This highly recommendable textbook is the companion volume of [N. Johnston, Introduction to linear and matrix algebra. Cham: Springer (2021; Zbl 1481.15001)]. It is indeed a second course in linear algebra.
This second part is divided in three main chapters: 1. Vector spaces; 2. Matrix decompositions; 3. Tensors and multilinearity.
Each chapter is divided into sections, summaries and reviews, and extra topics. These extra topics cover, for example, the QR decomposition, continuity and matrices, or semidefinite programming. Also, a brief review of introductory linear algebra is offered to the reader.
The book is well-organized. The main notions and results are well-presented, followed by a discussion and problems with detailed solutions. There are many helpful notes and examples. At the end of each section, the reader can frequently find several computational, true/false, or proof exercises. For many problems, one can find the solutions at the end of the book.
There are several illustrative and colorful figures. For instance, those illustrating the examples and remarks about the Gershgorin disc theorem or about the geometric interpretation of the positive semidefiniteness are really helpful.

MSC:

15-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra
15Axx Basic linear algebra
15Bxx Special matrices
47Axx General theory of linear operators

Citations:

Zbl 1481.15001
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