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Georgi E. Shilov
Publisher: Dover Publications (1977)
Details: 387 pages, Paperback
Topics: Linear Algebra
MAA Review[Reviewed by Allen Stenger, on 12/31/2010]
This is a very traditional course in the theory of linear algebra. It includes a detailed study of determinants, matrices, and quadratic forms, mixed with more abstract treatments of general linear spaces over a general field and of real and complex inner product spaces, and ending with chapters on algebras and on categories. Algebras are also used in the discussion of Jordan canonical form. The book is a translation by Richard A. Silverman of a Russian-language work; the prose is clear and easy to follow.
The book consists mostly of a narrative of definitions, theorems, and proofs, but is well-illustrated with many brief examples. There are a reasonable number of exercises, some numeric, but most asking for proofs and intended to continue subjects brought up in the text. The exercises are collected at the end each chapter rather than right after the section they apply to. All exercises have hints, and sometimes answers, in the back of the book.
The book largely ignores the practice of linear algebra. It does not deal at all with computational aspects such as LU-factorization or QR-factorization or even Gaussian elimination. There are no applications given, other than some applications to other areas of mathematics buried in the examples. The applications of linear algebra are so important today that omitting them is a serious drawback, and makes it difficult to motivate the development of the subject.
Bottom line: a competent. concrete, and easy-to-understand course in traditional linear algebra at a bargain price, but limited to the pure-math aspects and focused on concepts and proofs rather than techniques.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.