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kehler
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Homework Statement
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
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)