3 Advanced Linear Algebra Proofs
I am so bad at proofs.
1) Prove that if S is a subset of a vector space V, then span(S) equals the intersection of all subspaces of V that contain S.
Ok, so from the Theorem, I know that since S is a subset of V, the span of S is a subspace of V.
The theorem states that a subspace containing S must also contain the span of S, but not the other way around. This is where I get stuck.
2) Prove that a subset W of a vector space V is a subspace of V if and only if span(W)=W.
If I wanted to show right to left, what can I conclude from the fact that span(W)=W? Or should I show it by contraposition?
3) Let S1 and S2 be subsets of a vector space V. Prove that span(S1 intersect S2) is a subset of span(S1) intersect span(S2).
I don't even know where to start on this one..
I know that this theorem can be used to help prove at least the first one.
Theorem: The span of any subset S of a vector space V is a subspace of V. Moreover, any subspace of V that contains S must also contain the span of S.