When $\operatorname{span} (S_1 \cap S_2) = \operatorname{span}(S_1) \cap \operatorname{span}(S_2)$holds

Question

Let $S_1$ and $S_2$ be subsets of a vector space. When does the equality $$\operatorname{span} (S_1 \cap S_2) = \operatorname{span}(S_1) \cap \operatorname{span}(S_2)$$ hold? I have found two sufficient conditions: They are vector spaces; one is a subset of the other. Any others?

But what is the necessary and sufficient condition? Can it be that there are no necessary conditions, or in other words we cannot necessarily characterize $S_1$ and $S_2$? What are some other problems where this situation arises?

Answer 1

We clearly have

$$ S_1\subseteq\mathrm{span}(S_1) \text{ and } S_2\subseteq\mathrm{span}(S_2) $$

Hence we get

$$S_1\cap S_2\subseteq\mathrm{span}(S_1)\cap\mathrm{span}(S_2)$$

and because the right side is a vector space we get

$$\mathrm{span}(S_1\cap S_2)\subseteq\mathrm{span}(S_1)\cap\mathrm{span}(S_2)$$

and we conclude that the equation holds if we have

$$\mathrm{span}(S_1)\cap\mathrm{span}(S_2)=\{0\}$$

Thus we have found another sufficient condition.

Answer 2

Here is a sufficient condition: $S_1\cup S_2$ is a set of linearly independent vectors.