(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let V be a vector space over the field K.

a) Let {[tex]W_{k}:\ 1\leq k \leq m[/tex]} be m subspaces of V, and let W be the intersection of these m subspaces. Prove that W is a subspace of V.

b) Let S be any set of vectors in V, and let W be the intersection of all subspaces of V which contains S (that is, x E W if and only if x lies in every subspace which contains S). Prove that W is the set of finite linear combinations of vectors from S.

2. Relevant equations

3. The attempt at a solution

a) I got this part so I will skip this. Part b is where I am stuck at. Just assume W is a subspace of V.

b) From what I understand, the question wants me to prove that W=span of S. I seriously don't know what to do. I tried to prove that any vectors that are NOT the span of S cannot be in W, but I didn't know where to go from there.

From a book I read, b) is actually a theorem. It says "W is the smallest subspace of V that contains S" but unfortunately it doesn't show any proofs for it.

I feel like I have missed something. Any hints?

Any help would be appreciated.

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# Homework Help: Vector space question

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