Not many books intended for math majors are accessible to everyone. The book Linear Algebra Done Right by Sheldon Axler is, however, delightfully comfortable to read. I am not sure why everyone recommends this as a second course in linear algebra. In hindsight, I feel like this could have been my first linear algebra book but perhaps I have recency bias.
Infinite-dimensional spaces have always seemed mysterious but there aren't many special things about them other than some rules from finite-dimensional spaces which break. Here's a few.
In general, neither injectivity nor surjectivity alone imply invertibility of linear maps. For finite-dimensional vector spaces, however, one is enough.
Only operators on finite complex vector spaces guarantee existence of eigenvalues. See 5.A.18 for an example.
Here is a list of few questions that I found interesting.
The book has a page numbered .