Intersection and Span Assume $S_{1}$ and $S_{2}$ are subsets of a vector space V.
It has already been proved that span $(S1 \cap S2)$ $\subseteq$ span $(S_{1}) \cap$ span $(S_{2})$ 
There seem to be many cases where span $(S1 \cap S2)$ $=$ span $(S_{1}) \cap$ span $(S_{2})$ but not many where span $(S1 \cap S2)$ $\not=$ span $(S_{1}) \cap$ span $(S_{2})$.
Please help me find an example.
Thanks.        

In $\mathbb{R}^2$, we have:
$$
S_1 = \{(1, 0), (0, 1)\} \\
S_2 = \{(2, 0), (0, 2)\}
$$
$S_1 \cap S_2 = \emptyset$ . Yet, both $S_1$ and $S_2$ span $\mathbb{R}^2$.
