Product Description

Determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, Euclidean spaces, unitary spaces, quadratic forms in Euclidean and unitary spaces, finite-dimensional space. Problems; hints and answers.

Product Details

• Amazon Sales Rank: #28936 in Books
• Published on: 1977-06-01
• Original language: Russian
• Number of items: 1
• Binding: Paperback
• 387 pages

Language Notes
Text: English, Russian (translation)

Customer Reviews

Excellent Linear Algebra Text
This is a solid book, but requires a degree of mathematical maturity. Like many of the Dover publications of translated Russian mathematical texts, the book is clearly written, with good proofs that are easy to follow, lots of useful examples, and solutions to problems are given at the end of the book.

Readers should note that the author is a noted Russian mathematician, a former professor of mathematics at Moscow University, one of great centres of mathematical research and teaching in the world. Shilov collaborated with many important mathematicians such as Kolmogorov and Gelfand. If you have read any of Kolmogorov or Gelfand's excellent Dover books, then the style of this book is very similar to those.

Outstanding Book
I find it ironic that my two favourite Linear Algebra texts are this book and the Axler, for they are exact opposites: Axler shuns determinants, and Shilov starts with them and builds much of his theory off them. However, there is no book I have found that has such a deep and clear exposition of determinants. The first chapter alone makes this book worth buying.

However, there's an incredible amount of material in this book, and the later chapters are just as valuable. This is a dense book, but it is fairly easy to read once you get used to the style. I would recommend it to anyone learning linear algebra for the first time, as well as to people who want a deeper understanding or a different perspective.

Like I said before, this book is particularly useful when combined with a complementary text such as Axler, which provides a completely different approach to the subject. This book may come across as a bit old-fashioned, and some might say the material is obsolete, but I believe that everything contained in the book is useful, if only to give the reader a deeper understanding of the why's and how's of linear algebra. And plus: you can't complain about the price!

Not that good
Ordinarily I would give this book three stars, but I feel the rave reviews must be offset further than that. The first chapter on determinants is very good, it gives you the why and the how of everything. Perhaps this is due to the somewhat concrete, computational nature of determinants, which favors Shilov's approach. Shilov retains this sort of computational orientation throughout the text, with very little attention paid to the visual/intuitive aspects of linear algebra. A particularly atrocious example of this is chapter 5 on coordinate transformations. He derives formulae for a multitude of different types of coordinate transformations without ever describing what the transformation accomplishes in intuitive terms. This can be troubling even for someone who already has a reasonable understanding of the subject, because it means that ultimately his presentation amounts to little more than a mere presentation of a lifeless formula, and you are left to determine for yourself what it amounts to. Indeed, all of the chapters, with the exception of the first, share in this general flavor. Computational in flavor, as opposed to conceptual or abstract, while at the same time weak in visual/intuitive content. To me this is not a winning combination, and has made for a rather miserable read. In all fairness, however, I should add that his coverage is pretty good, albeit a bit unorthodox as far as the order of presentation is concerned. You can learn linear algebra from this book, it just won't be that fun.