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.

1) Suppose is a subspace. Then by definition, W is closed under addition, scalar multiplication. Then in particular, all linear combinations of vectors in S are in W... Also note S is contained in span(S).

2) What is the definition of the span? It's the set of all linear combinations of the vectors, no? Then what can we say about sums? Scalar multiples?

3) Let . Then x is some linear combination of vectors in , so in particular is a linear combination of vectors in S_1 and a linear combination of vectors in S_2...