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Differentiable Linear Transformation

  1. Mar 22, 2013 #1
    1. The problem statement, all variables and given/known data

    Let V be the linear space of all real functions Differentiable on (0,1). If f is in V define g=T(f) to mean that g(t)=tf'(t) for all t in (0,1). Prove that every real λ is an eigenvalue for T, and determine the eigenfunctions corresponding to λ.


    2. Relevant equations



    3. The attempt at a solution

    All I know is that f'=λf and T(f)=λf in general. I tried substituting the variables, and I ended up with only t, which doesn't make sense.
     
    Last edited by a moderator: Mar 22, 2013
  2. jcsd
  3. Mar 22, 2013 #2

    Fredrik

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    Can you use the definitions of "eigenvalue", "eigenfunction" and T to explain what "f is an eigenfunction of T with eigenvalue λ" means?

    Edit: You have already given a partial answer for that by saying that Tf=λf. This is the part that follows from the definitions of eigenvalue and eigenfunction. So now you need to use the definition of T to explain what Tf=λf means.
     
    Last edited: Mar 22, 2013
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