Show that two subsets A and B of a vector space V generate the same subspace if and only if each vector in A is a linear combination of vectors in B and vice versa.
(a) if two subsets A and B of a vector space V generate the same subspace, then each vector in A is a lin. comb. of vectors in B.
I am thinking by contradiction: Suppose subsets A and B of a vector space V don't generate the same subspace and a in A is a lin. comb. of B.
From this, it seems trivial that A generates the same subspace. However, I could be wrong or I just don't know how to articulate it from here
Don't worry about answering the other direction. I am focusing on this one now.
Then for the other direction assuming the contrary would be the way to go? Since we are assuming the contrary, there exists an a in A s.t. . Therefore, the span A > span B which is a contradiction.