I just bought "Introduction to Calculus & Analysis - Volume 1" by Richard Courant. I am also looking for a good book on Linear Algebra. 1) Is the Courant book (I ordered it from the internet so I didn't get it yet) good? 2) What good books on Linear Algebra are there?
This should be moved to the Science Books Review sections. In any case, Courant is excellent. I have yet to find an introductory book at its level. I have one an only one complaint about Courant: he presents a "weak" version of the fundamental theorem of calculus, only considering continuous integrands. I think this is due to the unavailability of the mean value theorem of differential calculus at the time he presents the theorem (he develops the integral calculus before differentiation), which is usually used to prove the stronger version. For linear algebra, have a look at Shilov.
'Linear Algebra Done Right' by Sheldon Axler is a good book for a first course in abstract linear algebra.
Is the Linear Algebra book by Shilov meant for first year undergraduate? Or does it require prior knowledge of linear algebra?
It's hard for me to say. My class (Advanced Linear Algebra 1) uses Friedberg, which I find inferior to Shilov. It's difficult to jump head first into Shilov's book without any prior exposure to some concepts from algebra. Are you a HS student? Have you read through university math literature before?
Then I suppose Shilov is a bit too advanced. It's inexpensive though, so you, if you want to try it, go for it.