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Similarity and change of basis

  1. Jul 21, 2007 #1
    1. The problem statement, all variables and given/known data

    Consider the 2-dimensional complex vector space V of functions spanned by sin x and cos x. For a fixed real number α, define a linear operator T ≡ Tα on V by putting T(f(x)) = f(x + α). Find the matrices [T]B and [T]E of T relative to the bases B = {cos x, sin x} and E = {cos x + i sin x, cos x − i sin x}. Find an invertible matrix P implementing the similarity between [T ]B and [T]E :

    [tex][T]_B = P[T]_EP^{-1}[/tex]

    2. Relevant equations



    3. The attempt at a solution

    I found T_B and T_E and I think I'm probably right because their tr and det are equal. Sorry if I don't show all that work because it is long. My problem is to get P. I am not sure how to get this. Thanks for any suggestions.
     
  2. jcsd
  3. Jul 21, 2007 #2

    radou

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    Homework Helper

    Well, what do you know about transformation matrices between different bases?
     
  4. Jul 21, 2007 #3
    OK. Got it now.
     
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