Thread: Vector space polynomials

Hi,
I am trying to complete a question in preparation for my exams next month, but I am struggling to get my head around the whole concept of it.
Here it is...

Let N be a positive integer, and let Vn of R(X) be the vector space of polynomials with real coefficients and degree at most N. Consider the derivative D on Vn:
D : Vn Vn
f(x) f '(x) and the composite map D^(2)= D D : Vn Vn

(i) Show that D and D^(2) are linear
(ii) Compute the rank and nullity of the map D^(2)
(iii) Decide whether D is diagonizable

I would really appreciate any help for any part of this question!
Thanks in advance,
Katie