Let $\displaystyle V$ be a vector space and $\displaystyle S \subset V$ with the property that whenever $\displaystyle v_1,v_2,...,v_n \in S$ and $\displaystyle a_1{v_1}+a_2{v_2}+...+a_n{v_n} = 0$, then $\displaystyle a_1 = a_2 = ...=a_n = 0$. Prove that every vector in the span of $\displaystyle S$ can be uniquely written as a linear combination of vectors in $\displaystyle S$.