Linear Algebra Done Right by Sheldon Axler looks like an excellent textbook. If I ever end up teaching or tutoring undergraduate linear algebra again I think I'd try it as a text. The book is open access and is available at
https://linear.axler.net (I have no affiliation with the author; I just like the book!)
One thing I like about this book is its approach to eigenvalues and eigenvectors. Most linear algebra books present eigenvalues as roots of the "characteristic polynomial", which is built from the "determinant", which in turn has some formula defining it. These objects are rarely motivated geometrically, and so you're left with limited understanding of just what an eigenvalue is or why linear transformations on finite-dimensional vector spaces must have them. Axler avoids determinants till Chapter 9 of the book, focusing instead on linear operators. The fact that operators must have eigenvalues pops out of the observation that iterating an operator on a given non-zero starting vector results in a set of vectors that must eventually become linearly dependent. This fact also leads to the development of the characteristic polynomial; you can then come at the determinant from this, more geometric, perspective.
#math #teaching #LinearAlgebra