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Vector space, subspace, span

  1. Mar 11, 2010 #1
    The problem statement, all variables and given/known data

    Suppose V is a vector space with operations + and * (under the usual operations) and W = {w1, w2, ... , wn} is a subset of V with n vectors. Show Span{W} is a subspace of V.

    The attempt at a solution

    I know that to show a set is a subspace, we need to show closure under addition and multiplication. I don't where to go from there. Any suggestions?
  2. jcsd
  3. Mar 11, 2010 #2


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    Maybe start by reviewing what Span{W} means. Quote the definition in your next post, ok?
  4. Mar 11, 2010 #3
    Start with multiplication.

    Span W = c*a*w1+...+c*an*wn

    Does this exist in V?

    For addition, add Span W to Span R or whatever you want to call it.
  5. Mar 11, 2010 #4

    The span is basically the set of all linear combinations of the vectors w1, w2, ... , wn. So then, I can define some vector S and some vector T in terms of w's:

    S = c1*w1 + c2*w2 + ... + cn*wn

    T = k1*w1 + k2*w2 + ... + kn*wn

    I think I get it now. I can see how S + T will be closed, and some constant a*S will be closed.
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