这是一份ucl伦敦大学学院 math0047作业代写的成功案
For the derivative map $d / d x: \mathcal{P}{3} \rightarrow \mathcal{P}{3}$ given by
$$
a+b x+c x^{2}+d x^{3} \stackrel{d / d x}{\longmapsto} b+2 c x+3 d x^{2}
$$
the second power is the second derivative
$$
a+b x+c x^{2}+d x^{3} \stackrel{d^{2} / d x^{2}}{\longmapsto} 2 c+6 d x
$$
the third power is the third derivative
$$
a+b x+c x^{2}+d x^{3} \stackrel{d^{3} / d x^{3}}{\longmapsto} 6 d
$$
and any higher power is the zero map.
MATH0047 COURSE NOTES ：
The canonical form equals $\operatorname{Rep}{B, B}(m)=P M P^{-1}$, where $$ P^{-1}=\operatorname{Rep}{B, \mathcal{E}_{2}}(\text { id })=\left(\begin{array}{ll}
1 & 1 \
0 & 1
\end{array}\right) \quad P=\left(P^{-1}\right)^{-1}=\left(\begin{array}{cc}
1 & -1 \
0 & 1
\end{array}\right)
$$
and the verification of the matrix calculation is routine.
$$
\left(\begin{array}{cc}
1 & -1 \
0 & 1
\end{array}\right)\left(\begin{array}{ll}
1 & -1 \
1 & -1
\end{array}\right)\left(\begin{array}{ll}
1 & 1 \
0 & 1
\end{array}\right)=\left(\begin{array}{ll}
0 & 0 \
1 & 0
\end{array}\right)
$$