I found Linear Algebra by Friedberg, Insel and Spence to be excellent. Very clear, modern notation, great exercises. It's also what Tao lectured from in 115A: https://www.math.ucla.edu/~tao/resource/general/115a.3.02f/
Since it doesn't neglect computation, it's great as an introductory text.
After that, and some other math courses you can move on to Linear Algebra Done Right.
I ended up getting a math PhD so it must have done somethings right!