Linear Algebra

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 This book may come across as a bit old-fashioned, and some might say the material is obsolete, but I believe that everything contained in the book is useful, if only to give the reader a deeper understanding of the why's and how's of linear algebra.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!

This book has several good points. First, it is extremely affordable. Second, it covers all the typical topics in a typical undergraduate linear algebra course within the first 4 chapters or so. This makes it a great reference. I bought it as a supplement to my old linear algebra textbook, and it is great for that purpose. In addition, it continues on to more advanced topics which may not be covered at the undergraduate level; for example, the heavy emphasis on determinants and the more rigorous treatment of spaces, leading to affine spaces. I also have Shilov's book on real analysis, so I like the concise yet thorough manner of the author.

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.

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.

This is an Excellent book, and also a very good price for their quality (I will never understand the reason for which other books, bad some of them, reach prices so high presenting topics that are treated by other books with so very low price and that present the topics in excellent form as this one). Comparing this book with the Mirsky one, I encounter substantial differences, although the content characteristic of the linear algebra doesn't change. It treats the determinants in a single formula and once for all. Mirsky makes it's definition using the whole content of the Sylvester Theory, more enriched, more substantial. Shilov extends the theory of the determinant along the book, while Mirsky makes it in a single chapter dedicated to its study. Mirsky conserves along the whole book a classic presentation of the linear algebra, while Shilov tries to enrich the topic introducing elements of modern algrebra. Mirsky has more exercises at the end of each chapter and don't give us the answer, Shilov includes less, but it has answers to the exercises at the end of his book. Considering Mirsky my favorite one, I give a very special place to Shilov since he give us a wider panorama and you terminate locating the lineal Algebra in a wider context inside the mathematics (definition of Algebras and a very good introduction to the algebra of the polinomiales, introduction to the tensorial algebra etc). This book is for an intermediate level, for undergraduates, and I recommend it to anyone that wants to study the linear algebra having the security that the concepts will be very clear. Their reading is easy to continue and it is a good introduction to the abstract algebra.