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Math Help - R^4 subspace question

  1. #1
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    R^4 subspace question

    there is U,W,V subspaces of R^{4} and dimU=dimV=dimW=3
    prove that U\cap V\cap W\neq{0}
    how i tried to solve it:

    dim(U+V)=dimU+dimV- dim(U\cap V)
    because U+V is a subspace of R^{4}then dim(U+V)\leq4
    so dim(U\cap V)\geq10
    dim(W+(U\cap V))=dimW+dim(U\cap V)-dim(W\cap U\capV)
    because W+(U\cap V) is a subspace of R^{4}then
    dim(W+(U\cap V))\leq4
    by inputing the previos data i get
    dim(W\cap U\cap V)\geq17
    is it correct?
    Last edited by transgalactic; June 24th 2011 at 04:12 PM. Reason: vbnn
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: R^4 subspace question

    Quote Originally Posted by transgalactic View Post
    dim(W\cap U\cap V)\geq17 is it correct?
    Impossible, W\cap U\cap V\subset \mathbb{R}^4 so, \dim (W\cap U\cap V)\leq 4 .
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  3. #3
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    Re: R^4 subspace question

    indeed, we have 4 = dim(R^4) ≥ dim(U+V) = dim(U) + dim(V) - dim(U∩V) = 6 - dim(U∩V).

    this means that dim(U∩V) ≥ 6-4 = 2.

    now we also have 4 ≥ dim((U∩V)+W) = dim(U∩V) + dim(W) - dim((U∩V)∩W) ≥ 5 - dim(U∩V∩W).

    this means that dim(U∩V∩W) ≥ 5-4 = 1.
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  4. #4
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    Re: R^4 subspace question

    ok i understand now
    thanks
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