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Homework Help: Invariant Subspaces

  1. Apr 13, 2010 #1
    1. The problem statement, all variables and given/known data
    Let T be a linear operator on a vector space V and let W be a T-Invariant subspace of V. Prove that W is g(T)-invariant for any polynomial g(t).


    2. Relevant equations
    Cayley-Hamilton Theorem?


    3. The attempt at a solution
    Im not sure how to begin. Ok so g(t) is the characteristic polynomial of T. If W is a T-Invariant subspace of V, then [tex]\forall[/tex]v[tex]\epsilon[/tex]W, T(v) [tex]\epsilon[/tex] W

    So for any T with a characteristic polynomial g(t), W is still T-Invariant...not sure if im even leading into the right direction. Any help on getting going with this proof?
     
  2. jcsd
  3. Apr 13, 2010 #2

    Fredrik

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    Staff Emeritus
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    Gold Member

    Just have g(T) act on an arbitrary x in W, and show that the result is in W. This is much easier than you seem to be expecting.

    Why are you saying that g is the characteristic polynomial of T? You said that g was arbitrary in the problem statement.
     
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