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Linear Transformations

  1. Sep 14, 2008 #1
    1. The problem statement, all variables and given/known data
    Let V be the vector space of all functions f: R->R which can be differentiated arbitrarily many times.
    a)Let T:V->V be the linear transformation defined by T(f) = f'. Find the (real) eigenvalues and eigenvectors of T. More precisely, for each real eigenvalue describe the eigenspace of T corresponding to the eigenvalue. (Since the elements of the vector space are functions, some people would use the term eigenfunction instead of eigenvector
    b)Now let T:V->V be the linear transformation defined by T(f) = f''. Prove that every real number eigenvalue is an eigenvalue of T


    3. The attempt at a solution
    I don't really know where to start :S
    The eigenvectors should be functions f where
    T(f) = eigenvalue x f
    So, f' = eigenvalue x f
    What do I do from here? I can't expand it cos I don't know the form of f(x)
     
  2. jcsd
  3. Sep 14, 2008 #2

    Defennder

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    Homework Helper

    Well I can think of a class of functions for which f(n) = [tex]\lambda[/tex]f. Though I don't know if they are the only possible types of functions which satisfy this property.
     
  4. Sep 14, 2008 #3
    Would that be the exponential functions?
     
  5. Sep 14, 2008 #4

    HallsofIvy

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    Staff Emeritus
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    In other words, you need to solve the differential equations [itex]df/dx= \lambda f[/itex] and [itex]d^f/dx^2= \lambda f[/itex]. That shouldn't be too hard.
     
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