Vector space polynomials
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 http://upload.wikimedia.org/math/4/2...9b30f02246.png 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 http://upload.wikimedia.org/math/4/b...3eafbb4972.png Vn
f(x) http://upload.wikimedia.org/math/0/5...78bbf9bac6.png f '(x) and the composite map D^(2)= D http://upload.wikimedia.org/math/8/4...ed408f6a5c.png D : Vn http://upload.wikimedia.org/math/4/b...3eafbb4972.png 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,