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Math Help - Linear Transformations and Range

  1. #1
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    Linear Transformations and Range

    Hi, I am trying to prove the following equality

    Range(T*T) = Range(T*)
    where T is a linear transformation and * denotes the adjoint.

    I know I must first show that Range(T*T) \subset Range(T*) and vice versa.

    so, Let w exist in R(T*T), then there exists a v in vector space V s.t.
    T*T(v) = w.
    Then I draw a blank, what's next??? Or am I even starting it correctly?

    Thanks
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Linear Transformations and Range

    Quote Originally Posted by jnava View Post
    Hi, I am trying to prove the following equality

    Range(T*T) = Range(T*)
    where T is a linear transformation and * denotes the adjoint.

    I know I must first show that Range(T*T) \subset Range(T*) and vice versa.

    so, Let w exist in R(T*T), then there exists a v in vector space V s.t.
    T*T(v) = w.
    Then I draw a blank, what's next??? Or am I even starting it correctly?

    Thanks
    WEll, you get the inclusion you mentioned for free (why?)
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  3. #3
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    Re: Linear Transformations and Range

    Quote Originally Posted by Drexel28 View Post
    WEll, you get the inclusion you mentioned for free (why?)
    Because I can just use T(v) as a vector all on its own, so then T*(v) = w which implies that w is in the R(T*)

    Am I right?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Re: Linear Transformations and Range

    Quote Originally Posted by jnava View Post
    Because I can just use T(v) as a vector all on its own, so then T*(v) = w which implies that w is in the R(T*)

    Am I right?
    Yes! So how about the second part? It's slightly trickier.
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  5. #5
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    Re: Linear Transformations and Range

    Do I just use the same approach?

    Let w exist in R(T*), then there exists a v in vector space V s.t.
    T*(v) = w. Then I am guessing there is an adjoint property to get to T*T(v) = w.....??
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