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Math Help - Subspaces

  1. #1
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    Subspaces

    Let U and V be subspaces of a vector space W. Prove that their intersection UV is also a subspace of W.
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  2. #2
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    Get to work on that.

    It is trivial to observe that the elements in the intersection are contained in the Vector Space.

    You have three things to show:

    1) There is a zero vector.
    2) Closed for vector addition.
    3) Closed for scalar multiplication.

    Let's see how you do it.
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  3. #3
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    a*U(0) = a*U = 0
    b*V(0) = b*V = 0

    (U + V)(0) = U(0) + V(0) = 0 + 0 = 0

    ???
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  4. #4
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by pakman View Post
    a*U(0) = a*U = 0
    b*V(0) = b*V = 0

    (U + V)(0) = U(0) + V(0) = 0 + 0 = 0

    ???
    when he said that the zero vector is in the intersection, it doesn't mean that the zero vector will be used for the closure. to test for the closure, you must pick arbitrary elements..
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