I think a lot of people taking courses in linear algebra lack a an adequate background in vectors, and this is detrimental to them. A couple of good possibilities for this would be "A Geometric Introduction to Linear Algebra" by Pedoe or "Analytic Geometry: With an Introduction to Vectors and Matrices" by Murdoch.

But if you do want to read a book on proofs, I'd suggest "Journey into Mathematics" by Rotman.