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.
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.”