Linear algebra proof - intersection of two subsets

$A,B$ are (finite) subsets in the vector space $V$ over the field $F$.

Prove or disprove the following:

$A\bigcap B= \emptyset \ \Rightarrow spA \bigcap sp B=\{ \vec0\}$

I understand that if both sets doesn't have any element in common then the only common vector would be the zero vector, otherwise they aren't subspaces of V but I don't really know where to start on how to prove it. Any help would be appreciated.

• What if $A=\{\vec{v}\}$ and $B=\{2\vec{v}\}$ for $\vec{v} \neq \vec{0}$? – user35603 Dec 4 '13 at 13:44
• I think you have to disprove it. – derivative Dec 4 '13 at 13:45

This is not true. Take $\Bbb R$ as $\Bbb R$-vector space and $A = \{1\}, B = \{2\}$. Then $\text{span } A = \Bbb R = \text{span } B$ over $\Bbb R$.