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Homework Help: Vector space question

  1. Jul 30, 2010 #1
    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.
     
  2. jcsd
  3. Jul 30, 2010 #2

    vela

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    Show W is contained in span(S), and span(S) is contained in W. Then W=span(S).
     
  4. Jul 30, 2010 #3
    Thanks for your reply vela.

    I get what you mean. You are saying if A is a subset of B and B is a subset of A, then A=B.

    Here is what I got so far:

    let S={[tex]\lambda_{1},\lambda_{2}...\lambda_{n}[/tex]}

    then, [tex]A1\lambda_{1},A2\lambda_{2}...An\lambda_{n}[/tex] E W1,W2,...Wm. (Closure under multiplication by a scalar.)

    and so [tex]A1\lambda_{1}+A2\lambda_{2}+...+An\lambda_{n}[/tex] E W1,W2,...Wm. (Closure under vector addition.)

    And so, span(S) E W1,W2...Wm

    Thus, span(S) is contained in W, as W is the intersection of W1,W2...Wm.

    How do I prove that W is contained in span(S)?

    Anyway thanks for your help.
     
  5. Jul 31, 2010 #4

    vela

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    Consider the fact that span(S) is a subspace of V that contains S.

    EDIT: I changed the wording in this post to say what I meant to say. Ignore what I had written here earlier.
     
    Last edited: Jul 31, 2010
  6. Jul 31, 2010 #5
    OMG I GOT IT.

    Since the span of S is a subspace of V, and W is the intersection of the subspaces in V that contains S, then obviously W E span{S}.

    Thus, span{S} E W and W E span{S} and so W=span{S}.

    Therefore, W is the set of finite linear combinations of S. =)

    Thanks so much man. You are awesome =D.
     
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