We are using Elementary Linear Algebra by Howard Anton in the class and I’m not happy with it. At times there is many pages of writing, yet, there is very little information contained. I really like vector spaces but to find appreciation of them I have to scavenge online sources. What would be a good supplement, maybe substitution? I don’t have knowledge beyond Calculus 3, and I really want to learn how to understand and write proofs. At the same time, I’d like to be able to follow at least half of the time what the author is trying to convey.
Linear Algebra Done Right by Sheldon Axler is a very clear book that takes a proof based approach to linear algebra and works natively with vector spaces with a minimum of algebraic matrix manipulation. I highly recommend it.
The book Linear Algebra by Friedberg, Insel and Spence is a very comprehensive book with lots of examples and exercises. You can find some examples of infinite dimensional vector spaces in the book. It even includes optional topics like dual vector spaces, which serves as a nice initiation to linear functional-analysis.
A completely different option would be to use the materials for the Massive Open Online Course titled "Linear Algebra: Foundations to Frontiers" on edX. It has online exercises, video, and simple programming exercises that links abstraction in mathematics to abstraction in programming. It is meant to be more appealing to students who are more computational science/computer science oriented.
This course does not require calculus and does include proofs of essentially all results.
I also recommend "Book of Proof" by Richard Hammack for instruction, examples, and exercises in writing proofs in general.
If you are a math major or studying to be a mathematician, and have already taken a course on matrix algebra, I recommend "The Linear Algebra a Beginning Graduate Student Ought to Know" by John Golan. Don't let the name fool you; while this book is aimed at a beginning graduate student, it's very accessible to undergraduates with a semester of basic vector-matrix algebra. This text will introduce you to some interesting ideas and notations that would supplement a theoretical linear algebra class well. It's not a tough read, and it will help you bridge the gap later when you move on to modern/abstract algebra courses.