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Math Help - trouble with inner product space problem...

  1. #1
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    trouble with inner product space problem...

    I am having trouble proving the following...
    Let B be a basis for a finite dimensional inner product space V.
    Prove that <x,z>=0 implies x=0 (the zero vector) for all z in B.
    Also prove that <x,z>=<y,z> implies x=y for all z in B.
    So far all I have is proof that at least one element of z must be nonzero, hence one element of x is zero, cant get any farther though.
    Any help would be appreciated.
    Thanks
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  2. #2
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    I think you need to learn to use quantifiers (read and write them) since what you wrote makes little sense as it is.

    Anyway from what I get you're trying to prove that if <x,z>=0 for all z\in V then x=0. If it's so then <z,z>= \Vert z \Vert ^2 =0 but a norm is zero iff z=0, for the second one use the first one with <x-y,z>=0 for all z
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  3. #3
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    I actually quantified z to be a vector in B, the basis for V
    what I did forget (thanks for pointing it out) is that x and y are vectors in V
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  4. #4
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    But the order is important, your quantifier should go before the implication (check it!), unless of course I understood something else in which case let me know.

    Anyway my proof assumes z was in V but it still works since if <,z> is zero on a basis of V then <,w>=0 for all w in V by bilinearity
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