Let us assume that every vector in $S_2$ is a linear combination of vectors in $S_1$.
Question: Does that mean that $S_1$ and $S_2$ are bases for the same subspace of $V$?
I know that the answer to this question is yes, subspaces spanned by both $S_1$ and $S_2$ are the same. Let's call them $W_1$ and $W_2$ respectively. How do we prove $W_1=W_2$? Equal subspaces when regarded as sets, must have the same elemnts. How can we show that EVERY vector in one subspace is also in the other subspace?