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Homework Help: Common complementary vector subspaces

  1. Sep 25, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that any two subspaces of the same dimension in a finite-dimensional vector space have a common complementary subspace. [You may wish to consider first the case where the subspaces have dimension 1 less than the space.]

    3. The attempt at a solution

    I've managed to sort out the case where the subspaces have dimension 1 less than the space I believe, using the first part of the question: "let U be a subspace of F^n. Show that there is a subset I of {1,2,...,n} for which the subspace W = span({e_i : i ∈ I}) is a complementary subspace to U in F^n." However, i tried using induction on the general case for k=dim(W) (so we've sorted the k=1 case if my proof is correct) and things got very messy and long. Could someone help me out with a more concise/neat solution which doesn't use too much complicated machinery?

    Thanks a lot :)
  2. jcsd
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