Linear algebra question on linear transformations. ? in thread. *Hard*

Question

anonymous · Answer

Let $n \ge 0$ be a natural number and $Pn$ denote the real vector space of polynomial functionsof degree up to $n$, i.e.

$$Pn(R) = \{f(x) = a_{n}x^n + a_{n-1}x^n-1 + ... + a_{1}x + a(0) |  a(i) \in R\}$$

Consider the linear transformation $D : Pn\rightarrow  Pn$ given by the derivative $D(f) = \frac{d}{dx}f$.

(a) Give a basis of the kernel of D
(b) What is the rank of D?
(c) Show that the n + 1-fold composition Dn+1 = D o  ... o D is the zero map.
(d) Show that the only eigenvalue of D is zero.
(e) Is D diagonalisable? Justify your argument.