If you haven't grasped the idea of a linear combination, I highly doubt you understand what a basis is.
What is it that you don't understand? Where do you get stuck reading the wikipedia article?
Prove that if V is a finite-dimensional vector space over a field F, then a subset { } of V is a basis for V over F if and only if every vector in V can be expressed uniquely as a linear combination of the .
First things first. Could someone explain the concept of a linear combination? Wikipedia and my book aren't quite cutting it.
What I've gathered from Wikipedia is that a linear combination is a set of scalars from a field multiplied by a set of vectors from a vector space. So in the literal sense of knowing the definition and knowing that a linear combination is the elements of a field multiplied by a the vectors in a vector space, I guess I understand it.
I don't understand what that does for us. And I have a rough thought of what a basis is, but probably not a well enough working idea to understand why that statement is true.