Chapter 5.9, Problem 8E

In Exercises $1-10$, the linear transformations $S,T,H$ are defined as follows:

$S:{\mathcal{P}}_3\to {\mathcal{P}}_4$ is defined by $S\left(p\right)=p^{\prime }\left(0\right)$.

$T:{\mathcal{P}}_3\to {\mathcal{P}}_4$ is defined by $T\left(p\right)=\left(x+2\right)p\left(x\right)$.

$H:{\mathcal{P}}_4\to {\mathcal{P}}_3$ is defined by $H\left(p\right)=p^{\prime }\left(x\right)+p\left(0\right)$.

Also, $B=\left\{1,x,x^2,x^3\right\}$ is the natural basis for ${\mathcal{P}}_3$ and $C=\left\{1,x,x^2,x^3,x^4\right\}$ is the natural basis for ${\mathcal{P}}_4$.

Let $p\left(x\right)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3$ be an arbitrary polynomial in ${\mathcal{P}}_3$.

a) Exhibit the coordinate vectors ${\left[p\right]}_B$ and ${\left[S\left(p\right)\right]}_C$.

b) If $P$ is the matrix for $S$ in Exercise $1$, demonstrate that $P{\left[p\right]}_B={\left[S\left(p\right)\right]}_C$.

1. Find the matrix for $S$ with respect to $B$ and $C$.