I dont know how to prove this although intuitively I know that it is true:
Let $ V $ be a finite dimensional vector space and $S$ and $T$ be subsets of $ V $.
Show that $$ Sp(S\cup T) = Sp(S)+Sp(T) $$ I think I have to show both sides are subsets of each other but I'm not sure how. I take it the question means sum of two vector spaces for the RHS
Edit: $Sp(S)=<S>$
$$S\subset \operatorname{Sp(S)}\subset \operatorname{Sp}(S)+\operatorname{Sp}(T),$$ and similarly $\,T\subset\operatorname{Sp}(S)+\operatorname{Sp}(T)$, whence $S\cup T \subset \operatorname{Sp}(S)+\operatorname{Sp}(T)$, and finally (remember the span of a subset is the intersection of all the subspaces which contain that subset): $$ \operatorname{Sp}(S\cup T) \subset \operatorname{Sp}(S)+\operatorname{Sp}(T).$$
Conversely, as $\,S\subset S\cup T$, $\,\operatorname{Sp}(S)\subset\operatorname{Sp}(S\cup T)$. The same is true for $T$, so that: $$\operatorname{Sp}(S)+\operatorname{Sp}(T) \subset\operatorname{Sp}(S\cup T).$$