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.

Finite v/s Infinite Vector Spaces

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.

Questions

Here is a list of few questions that I found interesting.