Linear algebra span proof

1. Mar 7, 2013

batballbat

1. The problem statement, all variables and given/known data
$$S_1$$ 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.

2. Relevant equations

3. The attempt at a solution
conjecture: iff the two subsets are vector spaces.

Last edited: Mar 7, 2013
2. Mar 7, 2013

Zondrina

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.

3. Mar 7, 2013

batballbat

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

4. Mar 7, 2013

HallsofIvy

Staff Emeritus
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?

5. Mar 7, 2013

batballbat

ok. plz delete this post.

6. Mar 7, 2013

Fredrik

Staff Emeritus
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?

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