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Linear Transformation Equalities

  1. Sep 9, 2011 #1
    The problem statement, all variables and given/known data

    I need to show the following equalities, where T is a linear transformation and * is the adjoint.

    Null(T*T) = Null(T)
    Range(T*T) = Range(T)



    The attempt at a solution

    I know I have to show that Null(T) is a subset of Null(T*T) and then Null(T*T) is a subset of Null T....

    Let v exist in Nullspace of T. Then T(v) = 0
    Now let v exist in the Nullspace of T*T. Then T*T(v) = 0

    Now i am lost, any help would be great. Thanks
     
  2. jcsd
  3. Sep 9, 2011 #2

    micromass

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    OK, so let v in the nullspace of T. Then T(v)=0. Can you show that T*T(0)=0??

    The reverse is not so easy. But take T*T(v)=0. Then <T*T(v),v>=0. Now use the property of the adjoint.
     
  4. Sep 12, 2011 #3
    Thank you!
     
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