1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Image of Normal Operator

  1. Mar 16, 2014 #1
    1. The problem statement, all variables and given/known data
    Show that if T is a normal operator on a finite dimensional vector space than it has the same image as its adjoint.

    2. Relevant equations

    3. The attempt at a solution
    I have been able to show that both T and [itex]T^{*}[/itex] have the same kernel. Thus, by using the finite dimension property and the rank nullity theoremit just suffices to show containment one way.

    However, if you suppose that a vector v is in the Im(T) I haven't been able to find some representative w such that T*(w) = v.

    Does anyone have any idea how to proceed?

  2. jcsd
  3. Mar 16, 2014 #2


    User Avatar
    Science Advisor
    Homework Helper

    You should also be able to show that Im(T) is orthogonal to Ker(T). Use that.
  4. Mar 16, 2014 #3
    Is that necessarily true? And if it is (I know that you can show that Im(T) is orthogonal to Ker(T*) and vice versa), but how does that specifically help?
  5. Mar 16, 2014 #4
    The idea is to show both ##\textrm{im}(T)## and ##\textrm{im}(T^*)## to be orthogonal to ##\textrm{ker}(T)##. For former you will need normality, the latter should be easy.
  6. Mar 16, 2014 #5
    I understand that but does that necessarily imply that they have to be equal? I recognize that they have to have the same dimensions in that case, but that doesn't get me anywhere.
  7. Mar 16, 2014 #6
    If you accept the fact in my last post to be true for now, then what would ##\textrm{ker}(T)^\bot## be?
  8. Mar 16, 2014 #7
    Ah it would be Im(T) and it would be Im(T*) so thus they have to be equal?
  9. Mar 16, 2014 #8
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted