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Linear Algebra (Dover Books on Mathematics)

38 customer reviews
ISBN-13: 978-0486635187
ISBN-10: 048663518X
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Editorial Reviews

Language Notes

Text: English, Russian (translation)
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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: 4.0 out of 5 stars  See all reviews (38 customer reviews)
  • Amazon Best Sellers Rank: #55,063 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

134 of 138 people found the following review helpful By S. Murphy on 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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75 of 76 people found the following review helpful By Alexander C. Zorach on 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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37 of 39 people found the following review helpful By Jason Dowd on 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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35 of 37 people found the following review helpful By Charles Saunders on 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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18 of 19 people found the following review helpful By mathwonk on March 2, 2008
Format: Paperback
I am choosing this book for my course on advanced linear algebra. This means nowadays a beginning course that covers all the bases, but that includes also some theory and proofs, and continues to the jordan form and spectral theorems.

I considered Axler, Lang, Hoffman Kunze, Halmos, and notes by Sharipov on the internet.

All these have their good points, but Shilov has it all: superbly clear explanations and proofs, examples and exercises, complete coverage of the important canonical forms, and a great elementary treatment of determinants, as well as tremendous attention to pedagogy.

E.g. like Halmos, Sharipov and some others, Shilov discusses nilpotent transformations separately and in detail, before doing jordan forms. since the idea of a jordan form is that every map is the direct sum of an invertible one and a nilpotent one, you would think it would make sense to discuss these types separately, but many books just cram the jordan form into one explanation with no discussion of nilpotent operators first.

finally, as a dover book, it is a terrific bargain. Friedberg Insel and Spence is a nice book, and Hoffman Kunze is also a classic, but those cost 10 times as much for about the same quality. I have reached the point in life where I will no longer assign a book that the publisher charges $135 for when there is a $15 book out there just as good or better.

Strangely however, not one student has ever expressed gratitude for this practice of mine in a class evaluation, but i suspect they appreciate it anyway, (or maybe Daddy is buying the books).

Edit: Having found cheap used copies of earlier editions of Friedberg et al..., I have relented and am using it also in my course. In general the earlier editions are better anyway. Some people have convinced me too that as clear as Shilov seems to me, it may be hard for some students to read.
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