This question is sort of related to my other one about diagonalizability. I actually learned something doing that previous question. Maybe someone can give me an explanation or a hint at first.

For each of the following linear operators T on a vector space V, test T for diagonalizability, and if T is diagonalizable, find a basis β for V such that [T]β is a diagonal matrix.

V = P2(R) and T is defined by T(ax2 + bx + c) = cx2 + bx + a.

I'm going to read some of section 5.2 about diagonalizability in my book but the book is abstract and hard to understand. I don't remember what I read about diagonalizability and need to go back a step.