OK. It's actually very easy when you know what you want to prove. Unless I'm mistaken of course.

Let . Suppose X spans the space. Then there must be a combination of vectors in X that is equal to Suppose Then since every vector in X has 1 on some coordinate different from . No combination of such vectors can be equal to . Therefore for any must be in X. If we suppose that X is independent, there can be no other vectors in it.