What Are the Eigenvalues and Eigenvectors of Derivative Linear Transformations?

[kehler]

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)

 

[Defennder]

Well I can think of a class of functions for which f⁽ⁿ⁾ = $$\lambda$$f. Though I don't know if they are the only possible types of functions which satisfy this property.

 

[kehler]

Would that be the exponential functions?

 

[HallsofIvy]

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