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Homework Help: Given a subspace S<=V, prove that there exists T<=V such that V=S⊕T.

  1. Oct 23, 2012 #1
    1. The problem statement, all variables and given/known data

    V is a vector space


    3. The attempt at a solution

    If S is smaller than V then there exists a T such that S + T = V. OTHERWISE S = V. I'm not sure what assumptions am I making which I could break down to prove...
     
  2. jcsd
  3. Oct 23, 2012 #2
    choose a basis A for T,complete it to a basis for V by adding a set B of vectors.Now show that the span of B is appropriate for T.
     
  4. Oct 23, 2012 #3
    How do I show that the span of B is appropriate for T?
     
  5. Oct 23, 2012 #4
    I've come up with this. Does this seem right?

    if S is the empty space, the solution is obvious
    if dim S >= 1 there is a base of vectors (ui) of S.
    And there is a theorem who says that
    the family of vectors (ui) can be completed
    with a family of vectors (vj) so that
    the union (ui) with (vj) is a basis of V
    finally the subspace T generated by (vj) = (v1, v2, ...)
    in the complementary space of S so that
    V=S⊕T
     
  6. Oct 23, 2012 #5
    i'm not sure if i need to quote the theorem
     
  7. Oct 23, 2012 #6
    Assume that v is a non zero vector in the intersection of S and T and prove that this contradicts the linear independence of the vectors in the union of A and B.
     
  8. Oct 23, 2012 #7
    How do I prove that? I don't see an obvious connection here
     
  9. Oct 23, 2012 #8

    Mark44

    Staff: Mentor

    hedipaldi,
    It is against Physics Forums rules to post complete solutions. You have received numerous warnings about this, and each comes with a private message to you. Apparently you aren't reading your PMs so I am posting something to you in this thread.
     
  10. Oct 23, 2012 #9
    I didn't ask for a complete solution, I'm genuinely stuck. I'm not that acquainted with the unusual rules here as I don't come here often. Each of your warning is about a separate issue and I don't repeat the same mistake again. I would appreciate if you understand my position.
     
  11. Oct 23, 2012 #10
    I know this rule and indeed i answered by hints.however it was not understood so i tried to help more.I understand that this is unwanted and i will obey the rules of the forum.
    sorry'
    Hedi
     
  12. Oct 23, 2012 #11
    Thanks for the help anyway :)
     
  13. Oct 23, 2012 #12

    Mark44

    Staff: Mentor

    My post was addressed to hedipaldi, not you, ashina14. See above.

    The rules are here: https://www.physicsforums.com/showthread.php?t=414380.
     
  14. Oct 23, 2012 #13

    Mark44

    Staff: Mentor

    I'm glad to hear that!
     
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