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6 orthonormal subspace question

  1. Oct 28, 2011 #1
    6)there is normal T in unitarian final space.
    [tex]v\neq0,v\in V[/tex] prove that if [tex]\{sp(v)\}^{\perp}[/tex] is T variant then
    v is eigenvector of T
    ?
    hint:prove that [tex]T*(v)[/tex] is orthogonal to [tex]\{sp(v)\}^{\perp}[/tex]
    what i have done:
    suppose [tex]u\in\{sp(v)\}^{\perp}[/tex]
    we take the definition of [tex]T*[/tex]
    (Tu,v)=(u,T*v)
    (Tu,v)=0 because [tex]\{sp(v)\}^{\perp}[/tex] is T variant so T*v is orthogonal
    to u.
    i know also
    V=[tex]\{sp(v)\}^{\perp}\oplus\{sp(v)\}[/tex]
    what to do next ,how to prove the actual question?
     
  2. jcsd
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