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Math Help - finite-dimensional subspaces

  1. #1
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    finite-dimensional subspaces

    Hi,
    Have been trying to understand linear Algebra for a month now. Am unable to solve this:
    If U,V,W are finite-dimensional subspaces of a real vector space, show that
    dim U + dim V + dim W - dim(U+V+W) >= max{dim(U intersection V), dim(V intersection W), dim(U intersection W)}

    Please help
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  2. #2
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    (dim U+dim W)+dim V-dim (U+V+W) ........(*)
    =dim (U+W)+dim (U\cap W)+dim V-dim (U+V+W)
    =dim (U\cap W)+dim ((U+W)\cap V)
    >=dim (U\cap W)

    Since U,V,W are independent, and (*) is symmetric, so you get three inequalities.
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  3. #3
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    Hi,
    What is capW?
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  4. #4
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    Quote Originally Posted by Sabita View Post
    Hi,
    What is capW?
    He means the following(though I dont know the validity of the results):

    Quote Originally Posted by KGene View Post
    (dim U+dim W)+dim V-dim (U+V+W) ........(*)
     =dim (U+W)+dim (U \cap W)+dim V-dim (U+V+W)
     =dim (U\cap W)+dim ((U+W)\cap V)
     >=dim (U\cap W)

    Since U,V,W are independent, and (*) is symmetric, so you get three inequalities.
    Actually I dont know what U+V means. Is it the direct sum?
    Like \mathbb{R}+\mathbb{R} = \mathbb{R}^2

    And the validity of his idea depends on one concept, dim(U \oplus V) = dim(U) + dim(V) - dim(U \cap V)
    But wiki says "The dimension of V ⊕ U is equal to the sum of the dimensions of V and U"...
    So I am not sure
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  5. #5
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    Quote Originally Posted by Isomorphism View Post
    He means the following(though I dont know the validity of the results):



    Actually I dont know what U+V means. Is it the direct sum?
    Like \mathbb{R}+\mathbb{R} = \mathbb{R}^2

    And the validity of his idea depends on one concept, dim(U \oplus V) = dim(U) + dim(V) - dim(U \cap V)
    But wiki says "The dimension of V ⊕ U is equal to the sum of the dimensions of V and U"...
    So I am not sure
    1. The sum U+V means the vector space consisting of elements of the form v+w where v in V and w in W, it is different from the direct sum U\oplus V. In the later case, we require  U\cap V=\emptyset if U,V are subspaces of a vector space. For instance, \mathbb{R}+\mathbb{R}=\mathbb{R} considering \mathbb{R} as a subspace of \mathbb{R}^2. But on the other hand, \mathbb{R}\oplus\mathbb{R}=\mathbb{R}^2.

    2. The "formula" dim(U \oplus V) = dim(U) + dim(V) - dim(U \cap V) was incorrect. The correct version is , dim(U + V) = dim(U) + dim(V) - dim(U \cap V). In general, dim (U\oplus V)=dim U+dim V

    3. In my first post, I meant the sum of two vector subspaces U+V, not the direct sum.

    Hope these help.
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  6. #6
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    Thanks KGene and Isomorphism.
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