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.