Could someone please explain to me the following?

Suppose you have a vector space, and a linear map T. need to find eigenvalues. when T is a matrix (2 by 2 or 3 by 3 or whichever square matrix with actual VALUES) this is easy, just work out the characteristic polynomial det(I*x - T)=0 and then the eigenvalues come out.

HOWEVER,

what does one do when T is a function? say, T(f) = f + f ' where f ' denotes the derivative of f?

so we don't have T as a matrix but as a function...

say, T is applied along the basis (1, x, x^2, x^3, ... , x^n)

do we need to apply T to every element of the basis and then get a matrix out of that to work with? or how??