Don't worry, Shilov is an excellent book. You really can't go wrong with it. I personally wouldn't recommend it because it does determinants in the beginning, which I find a fairly unintuitive and perhaps too abstract approach. Also, it never really says what a determinant is geometrically (as far as I recall). But if you like it, it's a nice book.Thank you micromass for the second insight this month that meets me where I'm at (the other was the one about advanced mathematics for high school students)!
As someone who is currently attempting to self-study linear algebra (albeit very slowly), this post helped me to see what I can expect to know once I'm done (by what micromass says the other books cover), especially since I'm using a fairly "mathematically pure" book to study from that doesn't give much motivation the whole subject.
Since it didn't make it on micromass' list, I'll go ahead and mention the book I'm using for anyone who is interested. It's by Shilov, just called "https://www.amazon.com/dp/048663518X/?tag=pfamazon01-20." It starts from determinants, which I hear is a different approach than most books take. The second chapter is on linear spaces though, so it seems that it still gets to "the point" rather quickly. I find it excellent, and perhaps it should be considered for someone who is looking into the subject.
Although I will point out that I have no prior experience with linear algebra and have looked at no other texts, so please take micromass' textbook advice over mine (if only this post had come out before I started to get into the book ).