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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?

user743574
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    Is this your first time learning linear algebra? Have you taken any proof-based math courses before? – Ben Grossmann Feb 09 '20 at 12:04
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    @Omnomnomnom Yes, this is my first time learning algebra. I have studied superficially proofs, not much. I'll start Calculus from Spivak, I need a book to start after Spivak. I know it will take a long time, but I want to buy them now, so I can have the road clear. – user743574 Feb 09 '20 at 12:07
  • Then I would recommend Shilov rather than Axler – Ben Grossmann Feb 09 '20 at 12:08
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    That being said, you might want to consider some of the choices from this list – Ben Grossmann Feb 09 '20 at 12:10
  • @Omnomnomnom Thanks for suggestion. Could you tell me which is more theoretical? I mean, the least pragmatic and more theoretical. – user743574 Feb 09 '20 at 12:14
  • A warning: I have read Axler's text, I have not read Shilov's. Going off of reputation alone, however, it seems clear that Axler's text is more theoretical, which is part of why I recommend Shilov instead. – Ben Grossmann Feb 09 '20 at 12:17
  • Get both, they are very different. Shilov gets to the point in the shortest possible way (even if not "elegant"), Axler strives to do everything in a mathematically pleasing way (modulo the motto "no determinants") Shilov does a lot of concrete calculations, Axler prefer abstract to concrete. – Artem Feb 09 '20 at 14:05
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    @Artem Interesting... What about Halmos' Finite Dimensional Vector Spaces? – user743574 Feb 09 '20 at 15:57
  • @Joãofodão I did not see this book. From all the suggestions provided below for my taste Hoffman and Kunze is the better option. – Artem Feb 10 '20 at 18:13
  • You don't have to commit to a single book. Read multiple books simultaneously and focus on whichever one is connecting with you. – littleO Nov 16 '20 at 04:19

2 Answers2

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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.

user729424
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  • The funny part is that a author, Serge Treil, wrote a book called Linear Algebra Done Wrong to make fun of Axler. This is exactly the reason why I didn't want Axler's book: the oversimplification. Ok, I didn't read it, but saw many people saying that he doesn't use something called "Determinants"(I don't even know what it is, but if it exists I want to learn it, and he wouldn't teach me it). The Hoffman and Kunze's book seems good for the reputation; however, I couldn't find it available (besides, it's expensive). I think I'm going to buy Halmos' and Shilov's books. – user743574 Feb 09 '20 at 16:53
  • If It's needed, I buy Axler's book after. For Abstract Algebra I was thinking about buying Jacobson's Basic Algebra I. Do you know this text? – user743574 Feb 09 '20 at 16:54
  • I've heard of Jacobson's books, but I'm not familiar with them. Algebra by Michael Artin is one of the most wonderful books in the whole world. – user729424 Feb 09 '20 at 16:57
  • It really makes me upset. It's like every book that people say are really great, isn't available for me. I didn't find Artin's Algebra available lol. – user743574 Feb 09 '20 at 17:00
  • Is the edition from Pearson India good? – user743574 Feb 09 '20 at 17:04
  • I don't know. I think I heard once that the Pearson India version of Algebra was missing the final chapter on Galois Theory. You should check to see if this is correct, because I may be mistaken. Also, please let me know if I'm wrong! If indeed it is missing that last chapter -- but it's not missing anything else -- then I'm sad, but I still think it's an excellent book. – user729424 Feb 09 '20 at 17:07
  • Probably I was who said it. I saw some contents in the Pearson India edition at Amazon saying that. Although I found the Pearson US edition that is complete (I guess) in second hand. I hate to buy book in second hand, so I think I'll buy the Jacobson Basic Algebra. Although I'll take a deeper look at it – user743574 Feb 09 '20 at 17:12
  • Yes, I remember now. We were discussing books on algebra. Nice to talk to you again! – user729424 Feb 09 '20 at 17:14
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    For me to you as well :) – user743574 Feb 09 '20 at 17:22
  • Axler does teach determinants, but introduces them only at a late stage. So in contrast with other courses, the subject is largely developed without using them. He thinks his proofs are clearer for not depending on this somewhat opaque concept. Also it's possibly the more general approach in that some of the ideas carry over to infinite-dimensional spaces. I do think it's unreasonable to call it an "oversimplification" for this reason. – Ali Feb 09 '20 at 20:26
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    The main problem with Axler is that he sells polynomial functions as "polynomials". Students come out unprepared for anything algebraic from this, and worse, the mis-definition, once learned, is hard to root out in subsequent classes. The treatment of determinants is comparably harmless, as I doubt anyone expects to get a full treatment of determinants from his book, and most American algebra textbooks bungle determinants in one way or another by not trusting the reader with any real mathematics. – darij grinberg Feb 09 '20 at 21:16
  • Fun exercise: Why does Definition 10.39 in Axler's own Measure, Integration & Real Analysis tacitly rely on polynomials being sequences of coefficients rather than functions $\mathbb{R}\to\mathbb{R}$ ? (To be fair, Axler proves that the coefficients of a polynomials are well-defined in LADR, but this proof is an afterthought rather than part of the definition, and suited only to $\mathbb{R}$ and $\mathbb{C}$. I wouldn't even know how to adapt it to multivariate polynomials!) – darij grinberg Feb 09 '20 at 21:18
  • @darijgrinberg I should have looked at Axler more closely before I made a comment about how Axler handles polynomials. I'm going to edit my post in response to your comment. Thank you for helping me make a better post! – user729424 Feb 09 '20 at 21:20
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    My advice is to read the foreward / intros by the authors themselves. Axler recommends his book for a second course in linear algebra not a first course-- conclusion it is not for the OP. Artin's 'note for the teacher' in the 1st edition at least, in essence says the book is way too much for even a 1 year course unless the students have significant mathematical maturity and already had 'a semester of algebra, linear algebra for instance.' Again this is not the OP. – user8675309 Feb 09 '20 at 22:19
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    One correction to the answer above: The statement that my book "defines a field to be either $\mathbf{R}$ or $\mathbf{C}$" is not correct. My book considers only vector spaces over $\mathbf{R}$ or $\mathbf{C}$ because I did not want to add the additional abstraction of arbitrary fields to the difficulty of learning abstract linear algebra. See the note titled "Digression on Fields" on page 10 of my book (third edition), where I state that many of the results in the book are valid over arbitrary fields. – Sheldon Axler Nov 14 '20 at 03:17
  • @SheldonAxler I apologize for the mistake that I made in my post. I've gone back and looked at my copy of your book (it's the second edition), and I realize now that you're right, and that I had indeed made a mistake. I've added another edit to my post to point out that I had made a mistake. I hope you don't feel that I've been unfair to you or to your book. Thank you for taking the time to read my post. – user729424 Nov 16 '20 at 03:53
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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.

LeonTheProfessional
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