If a set of vectors is a linear combination of another set, do they have the same span

Question

I'm trying to think about how to go about proving whether or not some list of vectors list has the same span as another (in general). For example, to prove something is a vector subspace, we need to prove the three axioms; but to prove vector $v$ has the same span as vector $u$, what must I show –that one is a linear combination of the other?

Answer 1

If you have two sets of vectors, $A$ and $B$, and if every vector in $A$ is a linear combination of vectors in $B$, then the span of $A$ is a subspace of the span of $B$.