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?