Originally Posted by

**dwsmith** 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.

$\displaystyle \mathbf{a}_i=\sum_{i\in I}\lambda_i\mathbf{b}_i, \ \lambda_i\in\mathbb{F}$

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 focusing on this one now.