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Linear Algebra (Dover Books on Mathematics) [Paperback]

Georgi E. Shilov
4.1 out of 5 stars  See all reviews (34 customer reviews)

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Book Description

June 1, 1977 048663518X 978-0486635187
Covers 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 with hints and answers.

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Linear Algebra (Dover Books on Mathematics) + Ordinary Differential Equations (Dover Books on Mathematics) + Fourier Series (Dover Books on Mathematics)
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Editorial Reviews

Language Notes

Text: English, Russian (translation)

Product Details

  • Series: Dover Books on Mathematics
  • Paperback: 400 pages
  • Publisher: Dover Publications (June 1, 1977)
  • Language: English
  • ISBN-10: 048663518X
  • ISBN-13: 978-0486635187
  • Product Dimensions: 8.4 x 5.6 x 0.8 inches
  • Shipping Weight: 11.2 ounces (View shipping rates and policies)
  • Average Customer Review: 4.1 out of 5 stars  See all reviews (34 customer reviews)
  • Amazon Best Sellers Rank: #45,191 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
125 of 129 people found the following review helpful
5.0 out of 5 stars Excellent Linear Algebra Text October 5, 2002
Format:Paperback
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.
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72 of 73 people found the following review helpful
5.0 out of 5 stars Outstanding Book December 10, 2005
Format:Paperback
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!
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34 of 35 people found the following review helpful
5.0 out of 5 stars Excellent Presentation of Linear Algebra May 21, 2003
Format:Paperback
Like most of the Dover books, this is a reprint of a classic text. This means that there is not a lot of hand-holding, only solid, clearly explained mathematics for those who have the motivation and want to put in the effort - not like our friend rururu who learns his math from Schaum's outlines. Shilov is one of the great mathematicians of the 20th century and so his proofs are well done and are very helpful for anyone wanting to have a real understanding of linear algebra and the follow on courses of ordinary and partial differential equations.
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28 of 29 people found the following review helpful
5.0 out of 5 stars An Excellent Second Book on the Subject October 16, 2011
Format:Paperback
If you are trying to learn linear algebra for the first time, avoid this book. You will be better served by Schaum's Outline of Linear Algebra Fourth Edition (Schaum's Outline Series). However, if you already have a working knowledge of the basics, have realized the central importance of this subject, and are trying to reach a deeper understanding of it, this is the book for you.

Here is a brief rundown of the contents:

Chapter 1 is on the theory of determinants. It is excellent and forms the basis of the entire book. It also immediately shows the sole focus of this book: finite dimensional linear algebra.

Chapter 2 is on linear spaces. Here as in his other books, Shilov moves to the abstract very early in his treatment. Anyone reading this book should be prepared by at least having the background of Schaum's Outline of Set Theory and Related Topics. It is noteworthy that this book does not even admit the existence of infinite dimensional vector spaces. Indeed, a basis for a vector space must be finite by definition. This is quite nonstandard, but again shows the focus: finite dimensional linear algebra.

Chapter 3 treats systems of linear equations. This chapter moves us back to the concrete, but again this book is only for readers with a firm grasp of the basics. Gaussian elimination is never even mentioned, but Cramer's rule is discussed in detail and used frequently for the rest of the book as a powerful theoretical tool.
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19 of 20 people found the following review helpful
Format:Paperback|Verified Purchase
5 stars for the author, but regrettably only one star for the translator. Dr. Richard Silverman should stay well within his expertise which is mathematical translation. Please do not second guess the thinking behind the Russian genius. Dr. Silverman, in the process of freely editing, transformed a masterpiece from a mathematical authority into a monster's piece. There are numerous typos and simple algebra, and sometimes conceptual errors to make the book legitimate for learning mathematics. For example, in section 6.1, the lengthy construction of the "Canonical Form of the Matrix of a Nilpotent Operator", The translator completely mixed up the construction of space (H sub r) making it almost impossible to follow. It is not a typo error as claimed by the translator.

However, regarding to the real content of the book,
it is a masterpiece from a mathematical genius. Brilliance mathematical expositions and beautiful proofs. The expositions are coherent and rigorous. And though, it would have been very readable if the mistakes were corrected.
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Most Recent Customer Reviews
2.0 out of 5 stars Review for the non-mathematician
As far from a practical linear algebra book as possible. If proofs and rigor aren't of your interests and reasons for learning LA, don't but this book. Read more
Published 1 month ago by Nate
1.0 out of 5 stars Linear Algebra
I bought the book to add to my information on Linear algebra. Didn't give me the information I was looking for.
Published 4 months ago by Wrinkles
5.0 out of 5 stars agree with other 5 stars: aside from a few missing "modern" features,...
The exposition, problems and graphics are top notch. The first chapter is Determinants, but this isn't one of those memorization and recipe type books for *applied" LA... Read more
Published 5 months ago by pounding on the keyboard
4.0 out of 5 stars clear and complete
The text is very clear and complete in explaining the background of the various tools.
At the other hand, the material is by no means easy. Read more
Published 15 months ago by Otto de Vries
5.0 out of 5 stars Great and smooth text.
Great text on the subject! Emphasis on proofs. Very interesting and explainable covering of determinants. Read more
Published 15 months ago by Walther
2.0 out of 5 stars If you find e-book
I chose this because I can read this using kindle.(I read only e-books)
But I think, if you find e-book about linear algebra, Kenneth's is better choice. It's free. Read more
Published 16 months ago by student
5.0 out of 5 stars Super easy to read
I am an engineering student who found the need to really learn Linear Algebra in order to move forward in my studies in stochastics and systems. This book is pretty awesome. Read more
Published 16 months ago by freefood
2.0 out of 5 stars not good
i was expecting something understandable. If you are interested to learn Linear Algebra don't start with this book, because it is very hard to follow.
Published 19 months ago by levani
4.0 out of 5 stars Price Says it All, but don't use alone!
For a little over 10 bucks, it's worth getting this VERY DENSE, terse Russian - translated version of LA. Read more
Published 22 months ago by Let's Compare Options
5.0 out of 5 stars An amazing reference in addition to a linear algebra textbook
When I first picked up this book, with a small mathematical background, it was tedious and overbearing because of the complex and generalized notation. Read more
Published 23 months ago by canyonclimber
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