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Linear Algebra (Dover Books on Mathematics)

4.1 out of 5 stars 44 customer reviews
ISBN-13: 978-0486635187
ISBN-10: 048663518X
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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: 5.6 x 0.8 x 8.4 inches
  • Shipping Weight: 11.2 ounces (View shipping rates and policies)
  • Average Customer Review: 4.1 out of 5 stars  See all reviews (44 customer reviews)
  • Amazon Best Sellers Rank: #209,915 in Books (See Top 100 in Books)

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Customer Reviews

Top Customer Reviews

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 This book may come across as a bit old-fashioned, and some might say the material is obsolete, but I believe that everything contained in the book is useful, if only to give the reader a deeper understanding of the why's and how's of linear algebra.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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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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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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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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Format: Paperback
Full disclosure: I'm not finished with this book yet. My last encounter with linear algebra was in 1978, so I expected some head-scratching. But it took me many hours just to get 25 pages into this book. Linear algebra is not Einstein science, so I was worried. I compared this book to my 30-year-old college textbook, and the college textbook wins. This book is lacking in examples. A theorem is presented, then proved, then given a reference number, and then pops up later by reference number, forcing use of many bookmarks.

On the positive side, the price is a bargain. And, the translation from the original Russian is absolutely invisible.
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