I think the book you're talking about is Axler's "Linear Algebra Done Right." I worked through the problems in this book with some friends, and it is a very good presentation.

After doing that book I wanted to see what other books were out there, which ones were the good ones, and sort of classify them. I came up with this list: https://begriffs.com/posts/2016-07-24-best-linear-algebra-bo...

I classify the books as Generalist (like Axler's), Theoretical (starting from e.g. modules rather than vector spaces), Numerical (matrix normal forms and algorithms), Practice (books full of problems), and a few other categories.