Let V be a vector space over a field F, and let $\displaystyle S\subset S'$ be subsets of V.

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

b) Show that span(S) is a subset of Span(S').

c) Take $\displaystyle V = R^3 $ and give an example to show that it is possible that Span(S) = Span (S') even though $\displaystyle S \subset S'$ and $\displaystyle S \neq S' $.

d) Let U,W be subspaces of V. Prove that U+W is also a subspace of V.