# Multivariable calculus and linear algebra (without matrices/determinants)

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.

• Linear algebra without matrices? – ÍgjøgnumMeg Feb 5 '18 at 14:23
• 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 ! – Emilio Novati Feb 5 '18 at 14:33
• 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. – Bogdan C Feb 5 '18 at 14:42
• 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. – Omnomnomnom Feb 5 '18 at 14:54
• 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. – CyclotomicField Feb 5 '18 at 18:01