由于
\begin{equation*}
\varphi(1)=0,\ \varphi (x^{k}) =kx^{k-1}, 1\leq k\leq n-1,
\end{equation*}
所以\(\varphi(1),\varphi(x),\varphi(x^2),\ldots,\varphi (x^{n-1})\)在基\((1,x,\ldots ,x^{n-1})\)下的坐标分别为
\begin{equation*}
\begin{pmatrix}
0\\0\\\vdots\\0\\0
\end{pmatrix},\begin{pmatrix}
1\\0\\\vdots\\0\\0
\end{pmatrix},\begin{pmatrix}
0\\2\\\vdots\\0\\0
\end{pmatrix},\ldots,\begin{pmatrix}
0\\0\\\vdots\\n-1\\0
\end{pmatrix}.
\end{equation*}
因此\(\varphi\)在基\((1,x,\dots,x^{n-1})\)下的表示矩阵为
\begin{equation*}
A=\begin{pmatrix}
0 & 1 & 0 & \cdots & 0\\
0 & 0 & 2 & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & \cdots & n-1\\
0 & 0 & 0 & \cdots & 0
\end{pmatrix}.
\end{equation*}
对任意\(f(x)=a_0+a_1x+\cdots+a_{n-1}x^{n-1}\in V\)，有
\begin{align*}
\amp \varphi\left(f(x)\right) \\
= \amp (1,x,\dots,x^{n-1})A\begin{pmatrix}
a_0\\a_1\\\vdots\\a_{n-1}
\end{pmatrix} \\
= \amp(1,x,\dots,x^{n-1})\begin{pmatrix}
a_1\\2a_2\\\vdots\\(n-1)a_{n-1}\\0
\end{pmatrix} \\
= \amp a_1+2a_2x+\cdots +(n-1)a_{n-1}x^{n-2}.
\end{align*}
