If you want to see the matrix point of view done well, there's Linear Algebra Done Wrong: https://www.math.brown.edu/streil/papers/LADW/LADW.html https://www.ma

If you want to see the matrix point of view done well, there's Linear Algebra Done Wrong: https://www.math.brown.edu/streil/papers/LADW/LADW.html https://www.math.brown.edu/streil/papers/LADW/LADW.html. You can read a bit about the motivation for doing it that way on that website. The title is a reference to a somewhat well-known book, Linear Algebra Done Right, which avoids using determinants to develop the theory (resulting in a somewhat novel/cleaner presentation). It's unfortunately not freely available online (published by Springer – I would suspect most university students can get it freely through their library's website, however).