Linear algebra is the study of vector spaces. In an elementary (undergraduate) linear algebra course the course is not made abstract because it is assumed that this abstraction will confuse many students. Rather, the course concentrates mostly on the vector space because it is easier to visualize. I cannot think of any easy way to explain what a vector space is, one that you would find in a popular mathematics book, I can give an abstract definition but that would not make much sense and you probably seen it already. Basically a vector space is trying to generalize the same properties you find in : like a basis, linear independence, ... and so on.