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For a nonempty subset S of a vector space V , define span(S) as the set...

For a nonempty subset S of a vector space V , define span(S) as the set of all linear combinations of vectors in S.

(a) Prove that span(S) is a subspace of V .

(b) Prove that span(S) is the intersection of all subspaces that contain S, and con- clude that span(S) is the smallest subspace containing S. Hint: let W be the intersection of all subspaces containing S and show W = span(S).

(c) What is the smallest subspace containing the empty set ;?

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