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**tttcomrader** So, to find eigenvectors of T, I have $\displaystyle T(f)=f'= \lambda f $ So I need to find some polynomials f, are there any?

the standard ordered basis of V is $\displaystyle \{ 1,x,x^2 \} $

So the matrix $\displaystyle [T]_{ \beta } = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{pmatrix} $ Not quite; the 1 should be at the bottom of the middle column.

So the eigenvalue of T is 0,1. No, 0 is the only eigenvalue.

Well, then any polynomial to the first power would satisfy T, right?