Shilov or Axler for Linear Algebra?

Question

I'm looking for a book on linear algebra. I found that Axler's and Shilov's books have a good reputation. Which of them is better? Which is more complete and suitable for theoretical study?

Answer 1

This answer is limited by the fact that I am not familiar with Shilov's text. I'm sorry about that.

Axler's book defines a field to be either $\Bbb{R}$ or $\Bbb{C}$. Perhaps these are the only fields that an analyst might care about. In any event, I found this to be an annoying oversimplification. If you're planning any further work in algebra, then you're going to need to look at another book that does things properly.

This comment is perhaps not helpful, but I wanted to mention that I've always hated the title of Axler's book, Linear Algebra Done Right. Did Axler think that all those other authors had aimed to do Linear Algebra the wrong way?

Here are some alternatives to Axler:

1. I'm not familiar with the book, but I've heard a lot of good things about Finite Dimensional Vector Spaces by Paul Halmos.

2. A long time ago, I had my first theoretical glimpse of linear algebra by reading Linear Algebra by Kenneth Hoffman and Ray Kunze. I thought it was a good book.

3. Some people might strenuously object to the following suggestion, and the people that do so have a lot of good points to make. You should listen to them. Anyway, here's the suggestion: you could learn a more theoretical approach to linear algebra by learning abstract algebra. A great place to start would be the book Algebra by Michael Artin. This is a book about abstract algebra, but while's it's teaching you abstract algebra it will do an excellent job of teaching you the theoretical approach to linear algebra. In my opinion, the book does a much better job of doing Linear Algebra than Axler's book does. Note: skipping linear algebra and going directly to abstract algebra might not be good advice. Please be careful.

Edit: I thought I should say a bit more about Axler's book since the original question asked about that book and Shilov's book, and I'm only familiar with Axler's book. The main feature of Axler's book is that he waits until the very end to do determinants, so that he does as much linear algebra as he can without using determinants. There are pros and cons to using this point of view. As Axler points out, determinants are difficult and often defined without motivation. So using determinants to do things can sometimes obscure what's really going on. On the other hand, determinants are really useful and important, and determinants are one of the things that a good linear algebra course should teach you about. To learn more about Axler's point of view, you can have a look at this article he wrote for American Mathematical Monthly.

Edit: Sheldon Axler has taken the time to point out a mistake in what I had written above. I had said that Axler defines a field to be either $\Bbb{R}$ or $\Bbb{C}$. This is not correct. What he does is he uses the letter $\Bbb{F}$ to denote either $\Bbb{R}$ or $\Bbb{C}$. He mentions that many of the results hold when $\Bbb{F}$ is an arbitrary field, but he does not discuss fields further.

Answer 2

I see that the author of the accepted answer only owns Axler's book. For me the opposite is true - I only own Shilov's book.

If Axler's text delays the concept of determinants, Shilov does the opposite - determinants are the first concept examined in his book.

I can only talk about this part, since I got stuck: It is rigorous, and rather formal - it features a definition of determinants based on permutations - very helpful if you were previously only confronted with things like the Leibniz-Formula. It then continues with cofactors and minors, with the usual "Lemma, Definition, Theorem" - Style. I think this is great if you want a treatment without any surplus to slow you down - personally I am looking for a book with a little more commentary to accompany Shilov's text for my studies.