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Seeing as Axler is very reluctant to talk about determinants and generally avoids computations and playing around with algebra, I'd like to get a book that will serve as a companion to Axler's unorthodox approach.

I'm not terribly interested in real-world applications. I don't much mind them, but the maths is what matters most to me.

All things considered, the book should certainly be mathematically rigorous without being particularly advanced - a first course with some minor prior exposure is assumed here. However, the reader has a strong foundation in proofs. The book should serve as a counterpart to Axler's approach; the book should, among other things, make up for the topics Axler lacks in.

I'd love to hear your suggestions, especially from people who have experience with Axler. The target audience is a pure maths major. Again, applications aren't necessarily a deal-breaker, but I'm not interested in lots of trivial exercises that exist only to show the real-world value of linear algebra; that, I do consider a deal-breaker.

Ius Klesar
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    Linear Algebra Done Wrong? – Artem Feb 06 '16 at 06:00
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    I suppose Hoffman Kunze's Linear Algebra would be a decent companion. – caffeinemachine Feb 06 '16 at 06:01
  • @Artem Not bad, I will make sure to write that down. Though LA Done Wrong does seem rather one-sided, just like its Done Right counterpart. I was hoping for a more well-rounded companion. – Ius Klesar Feb 06 '16 at 06:06
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    "Linear Algebra" http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X/ref=sr_1_1?ie=UTF8&qid=1454738842&sr=8-1&keywords=shilov%20linear%20algebra by Shilov is a lovely book which gets surprisingly far and does determinants first, so you might like it as a counterpart to Axler's book. Of course, it also has the advantage of being a Dover book, which makes it easy on your wallet. – Alex Wertheim Feb 06 '16 at 06:09
  • @Alex Wertheim I love Dover books, and I heard many great things about Shilov's book. Unfortunately, I've also heard it said on many occasions that Shilov is quite an advanced text, and not suited for a first course in linear algebra. – Ius Klesar Feb 06 '16 at 06:25
  • @caffeinemachine Is it true that Hoffman and Kunze's book has a reputation for being somewhat advanced, maybe even too much for a first course on linear algebra? A good foundation in proof writing is assumed, but apart from that, not much else except for a course on calculus (Spivak). Would this book still be an accessible first course option given these constraints? – Ius Klesar Feb 06 '16 at 06:30
  • @Luke I don't see any reason why Hoffman/Kunze's text would be a problem to somebody who has already done calculus from Spivak. I think Hoffman's text greatly complements Axler's. – caffeinemachine Feb 06 '16 at 06:36

3 Answers3

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I think Linear Algebra by Friedberg, Insel, and Spence is a careful, clear, and very standard/orthodox treatment of Linear Algebra.

littleO
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  • How does Friedberg et al hold up with respect to applications? Is the amount of trivial, real-world applications kept within reasonable bounds? – Ius Klesar Feb 06 '16 at 06:33
  • Yes, this book certainly doesn't go overboard with applications. It does have a few nice applications, such as sections on special relativity, solving homogeneous linear systems of ODEs, and Markov chains. – littleO Feb 06 '16 at 07:03
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Prasolov's: Problems and Theorems in Linear Algebra proved a nice companion for me in my Linear Algebra class. It deals with determinants and matrices more than Axler does (which isn't saying much).

Alekos Robotis
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You can always use the standard book:

Linear Algebra -Serge Lang

It is a very standard book with lots of good exercises and examples.

Hope you will enjoy going through it.

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