Question

linear transformation question.*hard*

Answer 1

Let n >= 0 be a natural number and Pn denote the real vector space of polynomial functions
of 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) memeber of R}

Consider the linear transformation D : Pn -> Pn given by the derivative D(f) = d(f)/dx.

(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 ... D is the zero map.
(d) Show that the only eigenvalue of D is zero.
(e) Is D diagonalisable? Justify your argument.