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Math Help - Inner product space and linear transformation proof

  1. #1
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    Inner product space and linear transformation proof

    Hi guys.
    I need to prove the following:

    Let T:V\rightarrow V be a linear transformation.
    Let ImgT be the span of a set B, and V be the span of a set S.
    I need to prove that if for every u\in S,v\in B:<T(u),v>=0 then T=0.

    I'm not sure where to start, I need some guidance please.
    thanks in advanced!
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  2. #2
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    Re: Inner product space and linear transformation proof

    Hey Stormey.

    For your problem can you show that if u is non-zero (in general) then since <u,v> = 0 implies u = 0 or v = 0, then if v is not necessarily zero then u = 0.

    if u = Tx and x is not necessarily zero, then you need to show T = 0 but for all x.

    So the proof now becomes showing that T = 0 satisfies Tx = 0 for all valid vectors in x. You could try proof by contradiction by assuming that T is non-zero and find a value for some realization of x' where Tx = 0 but Tx' != 0 for x' != x. This will prove the statement that T must be zero in order for Tx = 0 for all valid x in the vector space.
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  3. #3
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    Re: Inner product space and linear transformation proof

    Hi chiro. thanks for the help.

    Quote Originally Posted by chiro View Post
    Hey Stormey.

    For your problem can you show that if u is non-zero (in general) then since <u,v> = 0 implies u = 0 or v = 0, then if v is not necessarily zero then u = 0.
    but that just means that u and v are orthogonal, it can't tell that either is 0.

    Quote Originally Posted by chiro View Post
    if u = Tx and x is not necessarily zero, then you need to show T = 0 but for all x.

    So the proof now becomes showing that T = 0 satisfies Tx = 0 for all valid vectors in x. You could try proof by contradiction by assuming that T is non-zero and find a value for some realization of x' where Tx = 0 but Tx' != 0 for x' != x. This will prove the statement that T must be zero in order for Tx = 0 for all valid x in the vector space.
    what's 'x' here? vector in V? in S?
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  4. #4
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    Re: Inner product space and linear transformation proof

    When I say <u,v> = 0 I mean when this holds for all possible v where v can vary. I should have been more clear on that.

    Also x is in set S.
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  5. #5
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    Re: Inner product space and linear transformation proof

    OK, thanks!
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