Can you clarify the first line after "proof"? I'm not sure if is meant to be , or what. Also, (and I'm going with a different interpretation here) it's entirely possible (in fact overwhelmingly likely) that every element of B is linearly independent (I would say "not in the span of") S. In that case, you will get a set that's too big to be a basis, at least in finite dimension. Even in infinite dimension, it won't be linearly independent.