1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Common complementary vector subspaces
  1. Sums of subspaces? (Replies: 0)

Loading...