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The Problem:

Let [tex]S_{1}[/tex] and [tex]S_{2}[/tex] be subsets of a vector space [tex]V[/tex]. Prove that [tex]span\left(S_{1}\cap S_{2}\right)\subseteq span\left(S_{1}\right)\cap span\left(S_{2}\right)[/tex].

Give an example in which [tex]span\left(S_{1}\cap S_{2}\right)[/tex] and [tex]span\left(S_{1}\right)\cap span\left(S_{2}\right)[/tex] 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!

Let [tex]S_{1}[/tex] and [tex]S_{2}[/tex] be subsets of a vector space [tex]V[/tex]. Prove that [tex]span\left(S_{1}\cap S_{2}\right)\subseteq span\left(S_{1}\right)\cap span\left(S_{2}\right)[/tex].

Give an example in which [tex]span\left(S_{1}\cap S_{2}\right)[/tex] and [tex]span\left(S_{1}\right)\cap span\left(S_{2}\right)[/tex] 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!

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