Chapter 5.9, Problem 16E

Let $S$ be the transformation in Exercise $14$, let the basis for $P_2$ be $B=\left\{1,x,x^2\right\}$, and let the basis for $P_3$ be $D=\left\{3,3x-x^2,3x^2,x^3\right\}$. Find the matrix for $S$. Let $S:P_2\to P_3$ be given by $S\left(P\right)=x^3{p}^{″}-x^2{p}^{\prime }+3p$. Find the matrix representation of $S$ with respect to the natural bases $B=\left\{1,x,x^2\right\}$ for $P_2$ and $C=\left\{1,x,x^2,x^3\right\}$ for $P_3$.