Conditions for Strict Inequality in Span Intersection

[Bashyboy]

Homework Statement

Let $S_1$ and $S_2$ be subsets of the vector space $V$. Prove that $span (S_1 \cap S_2) \subseteq span(S_1) \cap span(S_2)$. Given an examples of $S_1$ and $S_2$ for which equality holds and for which the inequality is strict.

Homework Equations

The Attempt at a Solution

I actually solved the problem written out in part 1--it was rather easy. But in the course of solving it, I wondered about the necessary and sufficient conditions for $span(S_1) \cap span(S_2) \subseteq span(S_1 \cap S_2)$. I tried discovering them on my own, but this proved rather difficult. Are there any necessary and sufficient conditions?

 

[fresh_42]

Homework Statement

Let $S_1$ and $S_2$ be subsets of the vector space $V$. Prove that $span (S_1 \cap S_2) \subseteq span(S_1) \cap span(S_2)$. Given an examples of $S_1$ and $S_2$ for which equality holds and for which the inequality is strict.

Homework Equations

The Attempt at a Solution

I actually solved the problem written out in part 1--it was rather easy. But in the course of solving it, I wondered about the necessary and sufficient conditions for $span(S_1) \cap span(S_2) \subseteq span(S_1 \cap S_2)$. I tried discovering them on my own, but this proved rather difficult. Are there any necessary and sufficient conditions?

Simply have a look on two lines in a (Euclidean) plane and discuss the possible cases.

 

[WWGD]

Sorry to nitpick, but of course you want to avoid $S_1=S_2$. And then you can do it analytically: If $x \in S_1 \cap S_2$ , then $Span(S_1) \cap Span(S_2)$ will contain $Span {x_1}$ , etc. Then try to find general form for element on $Span(S_1) \cap Span(S_2)$ that is not on the left.