Customer Reviews

Linear Algebra (Dover Books on Mathematics)

5 star:		(19)
4 star:		(3)
3 star:		(2)
2 star:		(1)
1 star:		(1)

 

 

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.

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 This book may come across as a bit old-fashioned, and some might say the material is obsolete, but I believe that everything contained in the book is useful, if only to give the reader a deeper understanding of the why's and how's of linear algebra.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

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 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.

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.

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.

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.

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.

You can't get much better than Shilov when it comes to elegance of presentation. And the material here is fairly comprehensive, running smoothly from elementary topics to advanced ones. Although I will not recommend this text to beginning readers on the subject, I will recommend it to most others as an essential, useful, and very readable work.

There are a lot of linear algebra-matrix theory books around, and for my next course I picked Shilov's book. (It is advanced undergraduate/beginning graduate level.) So what is special about this it?
It has a good mix of theory and computation. It has many exercises, and they help in the learning, I think, better than some of the books I looked at as well. Some of them have hints, and some answers, listed in the back. Processing matrices is a visual thing. Shilov's book includes lovely pictures of matrices, and their block components. This is a big help in teaching matrix products, inverses, determinants, diagonalizations, not to mention applications to geometry, conical sections, projective space, and tangent planes.
The beautiful pictures are many, are superb, are artistically done and include graphical illustrations, figures, surfaces and more. They may not be created with the latest software, and yet they are timeless, and better IMO than the glossy ones found in newer editions.
Another thing I like about Shilov's book is its use of multilinear algebra. It is motivated by differential geometry of course, but it is great for applications to for example invariant theory.
There is a number of great alternative choices, some priced in the $ 100 range; and yet I found that Shilov ($ 10) has an edge; even forgetting the price.
Sample of topics: Standard fare on linear algebra and matrices, systems of equations etc; vector spaces and their duals, inner product, linear transformations and their adjoints; canonical form, quadratic forms & extrema, unitary space, Jordan forms, and a nice selection of timeless applications. Review by Palle Jorgensen, December 2008.