When Do Span Intersections Equal Span of Intersections in Vector Spaces?

[batballbat]

Homework Statement

$$S_1$$ and $$S_2$$ are subsets of a vector space. When is this:$$span(S_1 \cap S_2) = span(S_1) \cap span(S_2)$$ true? Prove it.

Homework Equations

The Attempt at a Solution

conjecture: iff the two subsets are vector spaces.

 

[STEMucator]

Homework Statement

$S_1$ and $S_2$ are subsets of a vector space. When is this:$span(S_1 \cap S_2) = span(S_1) \cap span(S_2)$ true? Prove it.

Homework Equations

The Attempt at a Solution

conjecture: iff the two subsets are vector spaces.

It's true when both sides are subsets of each other.

Choose an arbitrary element in each set and show it belongs to the other set both ways.

 

[batballbat]

sorry, but that is of no help. I am asking for a condition and a proof for "iff".

 

[HallsofIvy]

Well, what do you know and what have you tried? Do you know what "span" means?

Or do you just want someone to do the problem for you?

 

[batballbat]

ok. please delete this post.

 

[Fredrik]

Homework Statement

$$S_1$$ and $$S_2$$ are subsets of a vector space. When is this:$$span(S_1 \cap S_2) = span(S_1) \cap span(S_2)$$ true? Prove it.

Homework Equations

The Attempt at a Solution

conjecture: iff the two subsets are vector spaces.

It's easy to see that your guess is wrong. Let $\{e_1,e_2\}$ be the standard basis of $\mathbb R^2$. Let $S_1=\{e_1\}$ and $S_2=\{e_1,e_2\}$. We have $$\operatorname{span}S_1\cap\operatorname{span} S_2 =\operatorname{span}(S_1\cap S_2)$$ but neither $S_1$ nor $S_2$ is a subspace.

You will have to put in some effort of your own if you want help with the problem. In particular, you should include the definition of "span". Is $\operatorname{span}\emptyset$ defined?