Linear Algebra (Dover Books on Mathematics)

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.

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. Brilliantly mathematical expositions and beautiful proofs. The expositions are coherent and rigorous. And though, it would have been very readable if the mistakes were corrected. Note: I changed the rating from 4 stars to 5 stars because I couldn't find another book which has more comprehensive presentations and clearer explanations on the subject starting from the beginner's to more advance topics. You should own the book if you would like to master linear algebra.

The first read through, I wasn't a big fan. It is very proof heavy without really any visuals. My Linear Algebra book in college was the opposite and I thought I preferred that. Ended up making it about halfway through before going back to my old book. I gave it another read though and I think I prefer this book now. The proofs are succinct and force you to visualize in your head instead of glancing at a graphic as you read the equations. It is good practice for reading any technical/white-papers where you have to visualize equations.

I left a review of this book several years ago and am completely rewriting it. I previously gave it 5 stars but I can't really justify that anymore and have to lower it to 4. I still really like it and would absolutely recommend it for a second course on linear algebra; as a first course it would be a complete disaster (something along the lines of Rudin's Principles of Mathematical Analysis as a first exposure to calculus for someone who's never heard of a limit before). There's a lot of material built up step-by-step and I found almost all of the proofs to be mostly very clear, provided of course that you've already been exposed to the basics and can deal with the terseness of the book (though Shilov relies pretty heavily on rather ugly equations with a lot of explicit sub/superscripts where matrix equations would be cleaner). There are the usual typos that occur in any math book (a decent amount of them, actually), but I haven't found any major ones that hinder understanding. The first chapter on determinants is amazing and covers just about all you need to know about them, although the definition of the determinant may be unfamiliar and no motivation is given for it at first. Later chapters shed some light on this, and anyway the reader will quickly realize that this strange definition has all the properties they're familiar with. Chapters 2-4 also cover mostly familiar material but possibly from a slightly different perspective, and no doubt some of the material will be new.The early sections of Chapter 5 are the first example of why I lowered the rating. He defines a transformation matrix between bases and then uses that to get a transformation matrix between coordinates, but he never actually says what that matrix does. It is NOT the matrix that multiplies the column vector consisting of the old coordinates to give you the column vector of the new coordinates. Now, he never actually claims that it is, so it's not really an error per se, but then what is this transformation matrix he defines on p.122? It's the transpose of the matrix that actually acts on the coordinates in the way described above, but why would we care about the transpose? It makes no sense to me.Another baffling choice he makes is in chapter 7 (top of p.197 to be precise) where he suddenly switches the convention of having the columns of the matrix of a linear transformation represent the images of the basis vectors to having the rows represent the images. WHY? Attention is drawn to this switch but no reason for it is given. This means that if you want to stick with the convention used earlier (and why wouldn't you? EVERY linear algebra book I've seen uses it), you have to swap a bunch of indices and swap certain instances of the words "row" and "column" throughout the chapter. Also in chapter 7 he uses the letter A to represent entirely different functions within the same problem. A could be a fixed bilinear form and an arbitrary linear transformation within the same sentence. Or A could be a bilinear form acting in K' and another bilinear form acting in another (although isomorphic) space K''. I know what he means and I can follow the proofs, which are ultimately the things that really matter, but there's no good reason for such sloppiness. A final criticism of chapter 7 is that not once in the entire chapter does he explicitly state that a bilinear form B(x,y) can be written as x^TBy, even though using this matrix form would make many of the proofs much cleaner.The rest of the book, of which I've gone through most but not all, is similar. The general progression of topics is interesting and logical, the proofs are understandable, and it's enjoyable to work through, but be prepared to write a lot of marginal notes and/or cross out and replace some words here and there.