- #1
sakodo
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Homework Statement
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.
Homework Equations
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.