[jacobolus]

Tools from linear algebra can be accessible and useful to many people who don’t want to (or are not yet prepared to) prove nontrivial theorems. Indeed, a book like Axler’s should probably be used in a second semester-long linear algebra course for typical undergraduates wanting to study abstract mathematics; a gentler more concrete introduction would probably be better for students without previous exposure to linear algebra or hard mathematical thinking. For engineers or others who want to use linear algebra in practical contexts, something like Boyd & Vandenberghe’s new book might be a better for a first (or even second) course than Axler’s book, https://web.stanford.edu/~boyd/vmls/

Elkies’s post is in the context of a course for very well prepared and motivated first-year undergraduate pure math students who are racing through the undergraduate curriculum because most of them intend to take graduate-level courses starting in their second year.

Those two audiences are very far apart.

[throwawaymath]

> Those two audiences are very far apart.

Yes, that's precisely why I said, "This isn't Math 55: compared to Rudin and Halmos, Axler is a very accessible introduction to linear algebra for those who are ready for linear algebra."

How do you propose to teach linear algebra beyond basic matrix operations and Gaussian elimination if you're not teaching any theory? You can take some disparate tools from linear algebra (just like you can with analysis to make calculus), but The presentation of learning the mechanical tools of linear algebra versus the theory of linear algebra is a false dichotomy. Axler's textbook is a very nice compromise that provides students an understanding of why things are the way they are while still teaching them how to work through the numerical motions of things. You need not go so far as reading Finite Dimensional Vector Spaces if you want to avoid theory, but you need enough of it to put the mechanical operations in some kind of context.

[jacobolus]

Personally I think that the undergraduate mathematics curriculum does a poor job of exposing people to examples and concrete situations before introducing new abstractions.

Students are often entirely unfamiliar with the context (problems, structures, goals, ...) for the new abstractions that are rained down on them, and end up treating their proofs as little exercises in symbol twiddling / pattern matching, without much understanding of what they are doing.

The undergraduate curriculum is put in this position because there is a lot of material to get through in not much time, and students are generally unprepared coming in. Ideally students would have a lot of exposure to basic material and lots of concrete examples starting in middle school or before, but that’s not where we are.

[throwawaymath]

I think we're in agreement on that point. In my experience most peoples' difficulty with higher mathematics comes from the tendency of elementary and high schools to push students along through grades without ensuring they've really mastered the material. Unfortunately most students come to hate math because they're introduced to ever more abstract and complex material when they haven't achieved a solid foundation to build upon. I don't see this artifact of our education system going away any time soon.

[g9yuayon]

Personally I used David Lay's Linear Algebra and Its Applications as my first linear algebra book. It's more formal than Gilbert Strang's famous text book, but less formal than Linear Algebra Done Right. It's emphasis on geometric intuition really struck home, particularly the discussion on coordinate systems, change of basis, and quadratic forms.