Proving Subset Equality in Vector Spaces: S1∩S2 ⊆ S1 ∧ S2

[kNYsJakE]

The Problem:
Let $$S_{1}$$ and $$S_{2}$$ be subsets of a vector space $$V$$. Prove that $$span\left(S_{1}\cap S_{2}\right)\subseteq span\left(S_{1}\right)\cap span\left(S_{2}\right)$$.
Give an example in which $$span\left(S_{1}\cap S_{2}\right)$$ and $$span\left(S_{1}\right)\cap span\left(S_{2}\right)$$ are equal and one in which they are unequal.

Solution:
I could do the proof, so that is not a problem. I found an example when they are equal to each other, but I can't think of an example that those two are not equal. It'd be nice if you could explain it in general case, but it is okay if you just give me an example. Please help me on this!

 

[g_edgar]

The Problem:
Let S_1 and S_2 be subsets of a vector space V. Prove that span(S_1$$\bigcap$$S_2)$$\subseteq$$Span(S_1)$$\bigcap$$Span(S_2). Give an example in which span(S_1$$\bigcap$$S_2) and span(S_1)$$\bigcap$$span(S_2) are equal and one in which the are unequal.

Solution:
I could do the proof, so that is not a problem. I found an example when they are equal to each other, but I can't think of an example that those two are not equal. It'd be nice if you could explain it in general case, but it is okay if you just give me an example. Please help me on this!

Can you find two sets S_1, S_2 so that their intersection is empty, but they each span the whole space V ?

By the way, doesn't $$S_1\cap S_2$$ look better than S_1$$\bigcap$$S_2 ?

 

[kNYsJakE]

The Problem:
Let $$S_{1}$$ and $$S_{2}$$ be subsets of a vector space $$V$$. Prove that $$span\left(S_{1}\bigcap S_{2}\right)\subseteq span\left(S_{1}\right)\bigcap span\left(S_{2}\right)$$.
Give an example in which $$span\left(S_{1}\bigcap S_{2}\right)$$ and $$span\left(S_{1}\right)\bigcap span\left(S_{2}\right)$$ are equal and one in which they are unequal.

Solution:
I could do the proof, so that is not a problem. I found an example when they are equal to each other, but I can't think of an example that those two are not equal. It'd be nice if you could explain it in general case, but it is okay if you just give me an example. Please help me on this!

test

 

[kNYsJakE]

The Problem:
Let $$S_{1}$$ and $$S_{2}$$ be subsets of a vector space $$V$$. Prove that $$span\left(S_{1}\cap S_{2}\right)\subseteq span\left(S_{1}\right)\cap span\left(S_{2}\right)$$.
Give an example in which $$span\left(S_{1}\cap S_{2}\right)$$ and $$span\left(S_{1}\right)\cap span\left(S_{2}\right)$$ are equal and one in which they are unequal.

Solution:
I could do the proof, so that is not a problem. I found an example when they are equal to each other, but I can't think of an example that those two are not equal. It'd be nice if you could explain it in general case, but it is okay if you just give me an example. Please help me on this!

fixed

 

[kNYsJakE]

Can you find two sets S_1, S_2 so that their intersection is empty, but they each span the whole space V ?

By the way, doesn't $$S_1\cap S_2$$ look better than S_1$$\bigcap$$S_2 ?

Yes. so that means $$span\left(S_1\cap S_2)$$ equal to a zero set $$\left\{0\right\}$$?

And since both $$span(s_1)$$ and $$span(s_2)$$ are subspaces of V, they both have a zero set as well, so $$span(s_1)\cap span(s_2)$$ also should be a zero set. That meas they are equal. I didn't get your answer... Sorry

 

[kNYsJakE]

Linear Algebra. spans!

Homework Statement

The Problem:
Let $$S_{1}$$ and $$S_{2}$$ be subsets of a vector space $$V$$. Prove that $$span\left(S_{1}\cap S_{2}\right)\subseteq span\left(S_{1}\right)\cap span\left(S_{2}\right)$$.
Give an example in which $$span\left(S_{1}\cap S_{2}\right)$$ and $$span\left(S_{1}\right)\cap span\left(S_{2}\right)$$ are equal and one in which they are unequal.

Homework Equations

Nothing.

The Attempt at a Solution

Solution:
I could do the proof, so that is not a problem. I found an example when they are equal to each other, but I can't think of an example that those two are not equal. It'd be nice if you could explain it in general case, but it is okay if you just give me an example. Please help me on this!

 

[g_edgar]

Yes. so that means $$span\left(S_1\cap S_2)$$ equal to a zero set $$\left\{0\right\}$$?

And since both $$span(s_1)$$ and $$span(s_2)$$ are subspaces of V, they both have a zero set as well, so $$span(s_1)\cap span(s_2)$$ also should be a zero set. That meas they are equal. I didn't get your answer... Sorry

Find $$S_1$$ and $$S_2$$ so that $$S_1 \cap S_2 = \emptyset$$, $$\text{span}(S_1) = V$$ and $$\text{span}(S_2) = V$$ .

 

[quasar987]

Take V=R, S_1={1}, S_2={2}. Then S_1 n S_2 is empty, so span(S_1 n S_2)=0. And span(S_1)=R=span(S_2) so span(S_1) n span(S_2)=R.

More generally, let S_1={v_1,...,v_k} be a basis for a subspace W of V and let S_2={u_1,...,u_k} be another basis for that subspace such that {v_1,...,v_k} n {u_1,...,u_k} is empty (for instance, u_i=2v_i). Then span(S_1 n S_2)=0, while span(S_1) n span(S_2)=W.

The idea behind the example is of course that given a subspace, there are many distinct basis for it.

 

[kNYsJakE]

Find $$S_1$$ and $$S_2$$ so that $$S_1 \cap S_2 = \emptyset$$, $$\text{span}(S_1) = V$$ and $$\text{span}(S_2) = V$$ .

Yeah. That's what i meant.

Well, thanks.

 

[kNYsJakE]

Take V=R, S_1={1}, S_2={2}. Then S_1 n S_2 is empty, so span(S_1 n S_2)=0. And span(S_1)=R=span(S_2) so span(S_1) n span(S_2)=R.

More generally, let S_1={v_1,...,v_k} be a basis for a subspace W of V and let S_2={u_1,...,u_k} be another basis for that subspace such that {v_1,...,v_k} n {u_1,...,u_k} is empty (for instance, u_i=2v_i). Then span(S_1 n S_2)=0, while span(S_1) n span(S_2)=W.

The idea behind the example is of course that given a subspace, there are many distinct basis for it.

Thank you! =)

 

[Fredrik]

I suggest that you consider subsets S₁ and S₂ of $\mathbb R^3$ that each contain exactly two vectors.

 

[Fredrik]

Don't post the same question in two forums.

 

[HallsofIvy]

I have merged the other thread with this one.