I really like "matrices are the coordinate form of linear transforms." In the LA class I took the professor made a pretty big deal out of that, first by defining "lexicographic matrix basis" so he could write out matrices as vectors and then talking about mapping between the three different ideas.

There's still stuff he said that I'm unpacking today... that was a dense class.

This is a great book. I also recommend Halmos "Finite Dimensional Vector Spaces". The typical way linear algebra is introduced does not present a matrix as a linear transformation first and foremost.

I'm reading Strang's linear algebra book and he teaches it in terms of combining columns and rows which I think is a lot clearer than explaining it in terms of linear equations.