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Math Help - Is this a SubSpace?

  1. #1
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    Apr 2009
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    Is this a SubSpace?

    S is the set of all points P such that vector AP is orthoganol to vector u = (-3,2,-3), where point A = (1,-2,3)
    Is S a subspace of R^3 ?

    ---- The answer is "NO" but I dont know why.... can someone please explain?


    ------- My Work, which is wrong...
    I tried to solve it symbolically by saying S = {(p1,p2,p3) | AP.u = 0 }
    and then checking if it was closed under vector Addition & scalar multiplication.

    Addition:
    Let a = AP #1
    Let b = AP #2

    We know
    a.u = 0
    b.u = 0

    So under addition
    (a + b).u = 0
    a.u + b.u = 0
    (0) + (0) = 0
    Therefore closed under addition.

    Multiplication:
    Let a = some arbitrary AP

    We know
    a.u = 0

    Then
    b = k.a, where k = some scalar
    b.u = 0
    (k.a).u = 0
    k.(a.u) = 0
    k.(0) = 0
    Therefore closed under scalar multiplication. Therefore it is a subspace.

    -------

    The above work is wrong. I'm not sure why. Can someone please explain? Thanks
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  2. #2
    Super Member Gamma's Avatar
    Joined
    Dec 2008
    From
    Iowa City, IA
    Posts
    517
    doesnt contain the origin.
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