Question

: Let L be the operator on P3 defined by L(p(x)) = xp'(x) + p''(x). If p(x) = a0 + a1 + a2(1+x^2), calculate L^n(p(x)).

Answer 1

More info/answers to earlier parts of question:
Matrix representation A with respect to the standard basis for polynomials = \[\left[\begin{matrix}0 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 2\end{matrix}\right]\]

Matrix representation B with respect to [1,x,1+x^2] = \[\left[\begin{matrix}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2\end{matrix}\right]\]

Transition matrix from [1,x,1+x^2] to standard = \[\left[\begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]\]