Linear Algebra - help please!

(i) Let P_11(R) be the vector space of polynomials of degree less than or equal 11 over the field R and let D : P_11(R) -> P_11(R) be the linear map given by differentiation. Write
down the least positive integer n for which D^(2n) = 0 on P_11(R)

(ii) By factorisation of the formal expressions D^(2n) -I, D^n - I or otherwise, show that the mapping
D^9 - D^6 + D^3 - I : P_11(R) -> P_11(R)
is invertible, and write down an expression for its inverse in terms of D.
(iii)Hence find the unique solution a € P_11(R) to the differential equation

(d^9)a/da^9 - (d^6)a/da^6 + (d^3)a/da^3 - a = -x^7 -x^4

Sorry this is not in the latex form I have seen used but I am not completely sure how to use it and I don't have much time!
The P_11(R) is P with a subscript 11 but I'm not sure how to do it and I've seen it done like this in the forums before.

Am I right in thinking that for (i) n=6?
and (ii) I've done something similar but I'm not sure how to go about it in this case.
and (iii) we have done inverse is integration, but not sure how to apply it to a differential equation.. :confused:

help please!!

Reply 1

SimonM

i) is correct

ii) have you factorise it? Please post some working

iii) Once you know that D^9 - D^6 + D^3 - I is invertible, you should know it's inverse...

Reply 2

lolololol

SimonM

i) is correct

ii) have you factorise it? Please post some working

iii) Once you know that D^9 - D^6 + D^3 - I is invertible, you should know it's inverse...