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Adjoint of a linear operator

  1. Feb 19, 2010 #1
    1. The problem statement, all variables and given/known data
    T is a linear operator on a finite dimensional vector space. then N(T*T)=N(T). the null space are equal.


    2. Relevant equations



    3. The attempt at a solution
    this is my method, but its does not work if dim(R(T))=0. i'm only concerned with showing
    N(T*T) [tex]\subseteq[/tex] N(T). let x beong to N(T*T) then <T*T(x),y>=0=<T(x),T(y)> for all y in the vector space. thus, if dim(R(T)) > 0 then there exists y such that T(y) is not equal to zero so T(x)=0.

    any other methods out there?
     
  2. jcsd
  3. Feb 19, 2010 #2
    okay i think i got it if dim(R(T))=0 then ofcourse x is in the null space of T.
     
  4. Feb 20, 2010 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    But that's only true for the trivial case, the 0 matrix.
     
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