Prove whenever subset is LI then all vecs in its span are unique

Hi. Can someone give me a clue on this extra credit problem (I really need it since I bolloxed up the first quiz) in my graduate linear algebra course.

Given: V is a vector space such that whenever v_i is an element of S (for all i=1..n) and such that the linear combination of those vectors (summation a_i * v_i) is the zero vector only if all a_i = 0 (in other words, S is linearly independent).

Prove: Every vector in span(S) can be uniquely written as a linear combination of S.