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Proof based on projections

  1. Jul 9, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose T ∈ L(V) and U is a subspace of V. Prove that U is
    invariant under T if and only if PUTPU = TPU.


    2. Relevant equations



    3. The attempt at a solution
    Consider u[tex]\in[/tex]U. Now let U be invariant under T. Now let PU project
    v to U so that PU(v)=u. Therefore TPU(v)=T(u). Now
    since T(u) is[tex]\in[/tex]U, PU should project T(u) back to U and
    by the definition of P^2=P, PU must be an identity operator since T(u) is in U,
    the space PU projects T(u) to, so PUTPU(v)=PUT(u)=T(u)
    which is equivalent to TPUv since TPUv =T(u) thus PUTPU=TPU.
     
    Last edited: Jul 9, 2009
  2. jcsd
  3. Jul 10, 2009 #2
    would like to know whether or not its right or wrong, thank you!:smile:
     
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