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**Aryth** I'm working on a practice final exam, and I'm stuck on a problem:

Let $\displaystyle T: \mathcal{P}_2 \to \mathcal{P}_2$ be a linear operator defined by $\displaystyle T(f) = f' + f''$. ($\displaystyle \mathcal{P}_2$ is the set of all polynomials of order $\displaystyle \leq 2$).

(Note that $\displaystyle T$ is corresponding to a matrix)

(1) Find the matrix $\displaystyle A$.

For this, I'm not sure if I need to use a basis as a reference or not. Is the matrix representation of $\displaystyle T$ supposed to operate on coordinates of $\displaystyle f$? Could I not just assume the standard basis for this, $\displaystyle B = \{1,x,x^2\}$, and use that to find $\displaystyle A$?

I imagine I'd be solving the problem $\displaystyle A[a \ b \ c]^T = [0 \ 2a \ (2a + b)]^T$ since the general polynomial of order two has the coordinates $\displaystyle [a \ b \ c]^T$ in $\displaystyle B$ and, after being operated on by $\displaystyle T$, has coordinates $\displaystyle [0 \ 2a \ (2a + b)]^T$.

Can you tell me if I'm on the right track?

(2) Is $\displaystyle A$ diagonalizable?

If I continue with the method from (1), then $\displaystyle A$ is NOT diagonalizable, but, of course, I'm not certain that my answer in (1) is correct or not.

(3) Find a basis, $\displaystyle B$ of $\displaystyle \mathcal{P}_2$ such that $\displaystyle [T]_B$ is a diagonal matrix.

In this context, is $\displaystyle [T]_B$ just my $\displaystyle A$ from part (1) just with a different basis? I figured that it would be, but, if that were the case, couldn't I just work backwards from (3) to (1) and everything is obvious?