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Polynomial Linear Transformation

  1. Mar 22, 2013 #1
    Let V be the linear space of all real polynomials p(x) of degree < n. If p ∈ V, define q = T(p) to mean that q(t) = p(t + 1) for all real t. Prove that T has only the eigenvalue 1. What are the eigenfunctions belonging to this eigenvalue?

    What I did was
    T(p)= (lamda) p = q (Lamda) p(t+1) = q(t) (Lamda) p(t+1) = p(t+1)
    (Lamda) = 1

    Eigenfunctions are nonzero constant polynomials.

    Is this right though?
     
  2. jcsd
  3. Mar 22, 2013 #2

    LCKurtz

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    That may be right, but you certainly haven't proved it. What you have to work with is$$
    p(t+1) = \lambda p(t)$$Surely ##\lambda = 1## and ##p(t) \equiv c## (a constant) satisfy that condition. But maybe some other ##\lambda## and ##p(t)## work too. You need to prove that ##\lambda## must equal ##1## and then show the only polynomials that work when ##\lambda =1## are constants.
     
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