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> $\newcommand{\sp}{\operatorname{sp}}$ Let $V$ be a vector space over $F$ field, and let $A,B$ be two different, disjoint, non-empty sets of vectors from $V$.
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> 
> Prove or disprove the following:
> If $A \cup B$ are linearly independent, $\sp(A) \cap \sp(B) = \{0\}$.
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It's an easy one to example over $\mathbb{R}^2$, because the $\text{span}$ of all $\mathbb{R}^2$ vectors has $(0,0)$ in it, but something keeps telling me that it might be disproved for other vector spaces.