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

Let

be a linear operator defined by

. (

is the set of all polynomials of order

).

(Note that

is corresponding to a matrix)

(1) Find the matrix

.

For this, I'm not sure if I need to use a basis as a reference or not. Is the matrix representation of

supposed to operate on coordinates of

? Could I not just assume the standard basis for this,

, and use that to find

?

I imagine I'd be solving the problem

since the general polynomial of order two has the coordinates

in

and, after being operated on by

, has coordinates

.

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

(2) Is

diagonalizable?

If I continue with the method from (1), then

is NOT diagonalizable, but, of course, I'm not certain that my answer in (1) is correct or not.

(3) Find a basis,

of

such that

is a diagonal matrix.

In this context, is

just my

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?