I'm in the middle of taking a linear algebra class and browsing through online libraries I stumbled upon a book called Linear Algebra Done Right (by Sheldon Axler). I haven't looked at it yet but it is said to cover everything my class does but without invoking matrices and determinants (until the very end)

This said, I've toyed a bit with multivariable calculus and iirc, analyzing a multivariable function requires the use of matrices and determinants (eg. Hessian matrix)

My question is the following. Has someone used Axler's approach applied to other fields? Or it's just theory with no real use outside of linear algebra.

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    $\begingroup$ Linear algebra without matrices? $\endgroup$ – ÍgjøgnumMeg Feb 5 '18 at 14:23
  • $\begingroup$ In the Sheldon Axler's book the matrices are introduced in chapter $3$ pag $48$ , when linear maps are defined. And from this point there are $200$ pages to the end of the book ! $\endgroup$ – Emilio Novati Feb 5 '18 at 14:33
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    $\begingroup$ There's a difference between mentioning them and profusely using them over the course of 200 pages. Of course he's going to mention them early on, the whole point of the book is that you can do without them. It makes sense that the author states this from the beginning. $\endgroup$ – Bogdan C Feb 5 '18 at 14:42
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    $\begingroup$ Axler's approach translates well to functional analysis, which is to say "infinite dimensional linear algebra". Functional analysis provides important underpinnings for PDE's and the math behind quantum mechanics. $\endgroup$ – Omnomnomnom Feb 5 '18 at 14:54
  • $\begingroup$ Axler's dogmatic approach has little application outside of religion. There is no reason to handicap your study of linear algebra in this way beyond faith-based claims. $\endgroup$ – CyclotomicField Feb 5 '18 at 18:01

Unfortunately this question and some of the comments contain unfair distortions. My book Linear Algebra Done Right introduces matrices in Chapter 3 and then frequently interprets results in terms of matrices throughout the following chapters. The emphasis in my book is on linear maps rather than on matrices, an emphasis that is appropriate for a second course in linear algebra. Nowhere in the book do I state or hint that “the whole point of the book is that you can do without them [matrices]”; thus that comment is simply wrong.

Although my book does not avoid matrices, it does leave determinants until the end in Chapter 10, showing that many results in linear algebra that are traditionally done with determinants can be done simpler and cleaner without determinants. My article in the American Mathematical Monthly presenting this viewpoint won the Lester Ford Award from the Mathematical Association of America. Thus I resent the assertion in one of the comments that this approach is based on “religion” and “faith-based claims”. I have carefully presented logical reasons why my approach has multiple advantages. I have no objection to folks who have a different opinion, but there is no need for name calling linking these arguments to “religion” and “faith-based claims.”

  • $\begingroup$ Thank you very much and sorry for the confusion I have caused (and for the weirdos it attracted), I will mark the question for deletion (unless you think it's worth anything), but your answer does indeed clear up a lot of things. I haven't read your book. This was a "before I buy" question. Just needed to be assured that this is a good investment (got to be careful with hard-earned student money). I will definitely give it a read now. Much appreciated! $\endgroup$ – Bogdan C Feb 7 '18 at 0:07
  • $\begingroup$ Bodgan: Thanks for your comment above. It's fine with me if you would like to delete the question. Whoever told you that the point of my book is that linear algebra can be done without matrices was giving you seriously incorrect information, as should be clear from looking at the book. (However, you can do linear algebra without determinants.) You might also like to look at the free videos (linear.axler.net/LADRvideos.html) I made to accompany my book; you will see matrices scattered throughout those videos. $\endgroup$ – Sheldon Axler Feb 7 '18 at 5:18

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