# 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 $(S_{2})$.

Thanks.

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HINT: Let $S_1=\{v\}$, where $v$ is a non-zero vector, and let $S_2=\{2v\}$.
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 $\mathbb{R}^2$.