Let {u1, . . . , um} be a set of vectors in an n-dimensional vector space

Question:

Let {u1, . . . , um} be a set of vectors in an n-dimensional vector space V and let B be a basis for V. Let S = {[u1]B , . . . , [um] B} be the set of coordinate vectors of {u1, . . . , um} with respect to B. Prove that span(u1, . . . , um) = V if and only if span(S) = Rn.