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Math Help - Finite-dimensional subspaces..?

  1. #1
    Super Member fardeen_gen's Avatar
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    Finite-dimensional subspaces..?

    Let U, V, W be finite-dimensional subspaces of a real vector space. Show dim U + dim V + dim W - dim (U + V + W) greater or equal to max {dim (U intersection V), dim (V intersection W), dim (U intersection W)}.

    - asked on behalf of a friend who is a non MHFer.

    I personally have no idea about vector spaces.
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  2. #2
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    Quote Originally Posted by fardeen_gen View Post
    Let U, V, W be finite-dimensional subspaces of a real vector space. Show dim U + dim V + dim W - dim (U + V + W) greater or equal to max {dim (U intersection V), dim (V intersection W), dim (U intersection W)}.

    - asked on behalf of a friend who is a non MHFer.

    I personally have no idea about vector spaces.
    i hope your friend is better than you at linear algebra! anyway, we have:

    \dim(U \cap V) + \dim ((U+V)\cap W)=\dim U+ \dim V + \dim W -\dim(U+V+W). thus: \dim(U \cap V) \leq \dim U + \dim V + \dim W -\dim(U+V+W), \ \ \ \ \ \ \ \ \ \ \ \ \ (1)

    and: \dim(U \cap W) + \dim ((U+W)\cap V)=\dim U+ \dim V + \dim W -\dim(U+V+W). thus: \dim(U \cap W) \leq \dim U + \dim V + \dim W -\dim(U+V+W), \ \ \ \ \ \ (2)

    and: \dim(V \cap W) + \dim ((V+W) \cap U)=\dim U+ \dim V + \dim W -\dim(U+V+W). thus: \dim(V \cap W) \leq \dim U + \dim V + \dim W -\dim(U+V+W). \ \ \ \ \ \ \ (3)

    now (1), (2), and (3) give us the result and, much more importantly, this is my 1000 post!
    Last edited by NonCommAlg; May 5th 2009 at 07:09 AM.
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  3. #3
    Super Member fardeen_gen's Avatar
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    Congratulations! Eventful journey on MHF indeed.

    P.S. My friend is 24 yrs old. I am a 16 yr old I have no idea what linear algebra is!
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  4. #4
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    Quote Originally Posted by fardeen_gen View Post

    P.S. My friend is 24 yrs old. I am a 16 yr old I have no idea what linear algebra is!
    wow, you're just a kid! i think it's quite late in India now. shouldn't you be in bed? (just kidding!)
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