$A,B$ are finite sets in a vector space $V$ over $F$.
Prove or disprove the following:
$\sp A \cap \sp B = \sp(A\cap B)$
$B\cap \sp A = \emptyset \Rightarrow A\cap \sp B= \emptyset$
For 1. the intersection of all the vectors that span $A$ and $B$ is the span of the intersection of $A$ and $B$. It seems so trivial that I find it difficult to formally prove...
As for 2. I couldn't find a counterexample but I'm not really sure on how to start.
Any help would be appreciated.