Define $\alpha$ as $\alpha(p)(x)=(x-1)(p'(x)+p'(1))$ Let V also be the space of polynomials of degree less than or equal to 2. The basis given is ${{1,x,x^{2}}}$
After after what I hope to have been a correct computation, I got the matrix $\mbox{} \left[\begin{array} 11 & x & x^{2} \\ 0 & 0 & 0 \\ -1 & -1 & -1 \end{array} \right]$
Now I can get the characteristic polynomial, to get the eigenvalues and then the nullspaces of each eigenvalue, if the number of distinct eigenvalues is equal to 3, then the matrix is diagonalizable, if not then I can find the Jordan canonical form.
Is this all correct?