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Math Help - need help proving a statment with subspaces dimensions

  1. #1
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    need help proving a statment with subspaces dimensions

    Hi.

    Let U,W be subspaces of V.
    Suppose dim(V)=n, and dim(U)=dim(W)=n-1
    I need to prove that dim(U\cap W)=n-2

    Thanks in advanced!
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  2. #2
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    Re: need help proving a statment with subspaces dimensions

    this is not always TRUE.

    if U = W, then dim(U) = dim(W) = n -1, but dim(U∩W) = n-1.

    you can't prove things that are not true.
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    Re: need help proving a statment with subspaces dimensions

    Deveno is right, your problem probably states dim(U\cap W)\geq n-2
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    Re: need help proving a statment with subspaces dimensions

    sorry, let me rephrase that:

    i need to prove that if dim(V)=n and dim(U)=dim(W)=n-1, then dim(U\cap W)=n-2.
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  5. #5
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    Re: need help proving a statment with subspaces dimensions

    that's not true. see post #2.
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    Re: need help proving a statment with subspaces dimensions

    Hi, Denovo.
    i just noticed it says there that W and U are different.

    sorry for the confusion.
    the mid exams are next week, so i don't sleep much these days...
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  7. #7
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    Re: need help proving a statment with subspaces dimensions

    then it's a different story.

    since W and U are distinct subspaces, say with bases B and B', respectively, then B' must contain a vector not in span(B). for if not, then U ⊆ W, in which case dim(U) = dim(W) forces U = W.

    hence dim(U+W) = dim(span(B U B')) = n (glossed over a bit: if we call this vector uj, with B = {w1,...,wn-1}, B' = {u1,...,un-1}, we need to show:

    C = BU{uj} is linearly independent. however, if:

    c1w1+...+cn-1wn-1+cnuj = 0, we can distinguish 2 cases:

    a) cn = 0. in this case, the linear independence of the wi forces c1 = ... = cn-1 = 0,

    b) cn ≠ 0. in this case, uj = (-c1/cn)w1+...+(-cn-1/cn)wn-1, contradicting our choice of uj.

    we also have n-1 = dim(span(B)) < dim(span(B U {uj}) ≤ dim(B U B') ≤ n).

    therefore: dim(U∩W) = dim(U) + dim(W) - dim(U+W) = n-1 + n-1 - n = 2n - 2 - n = n - 2.
    Thanks from Stormey
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  8. #8
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    Re: need help proving a statment with subspaces dimensions

    thanks again Deveno!
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    Re: need help proving a statment with subspaces dimensions

    but for the converse one? So far, I only have a result that if K is a set consisting of all nonunits of R, then for any u, v in K, u+v won't be 1R. Does it help? Any suggestion?





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