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Linear Transformation Equality

  1. Sep 12, 2011 #1
    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) 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
     
  2. jcsd
  3. Sep 13, 2011 #2

    lanedance

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    Homework Helper

    how about something like this as a strawman:

    first recognise that range is given by the span of the column vectors

    now consider T*u, for an arbitrary vector u. The results is a linear combination of the vectors in T*

    So the columns of T*T can be thought of as linear combinations of the columns of T*

    Then you probably need to invoke that the dimension of the column and row space are the same to finish
     
  4. Sep 13, 2011 #3
    I am slowly pounding through it. This is what I have:

    Let w exist in R(T*T), then there exists a v in V s.t.

    T*T(v) = w
    => T(v) exists in R(T*) => w exists in R(T*) => R(T*T) is a subset of R(T*)

    The other way is trickier
     
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