多项式乘以 `$x^2$` 为线性映射 定义如下线性映射: `$T \in L( P( R ), P( R ) ) $` 对于所有 `$x \in R$` , `$(Tp)(x) = x^2p(x)$` 此映射为一线性映射，证明过程如下 先来证明加性(additivity) 设

\begin{align*} u = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \end{align*}

\begin{align*} v = b_0 + b_1 x + b_2 x^2 + \cdots + b_n x^n \end{align*}

然后

\begin{align*} T(u + v) &= x^2 (u + v) \\ &= x^2 \Big((a_0 + b_0) + (a_1 + b_1) x + \cdots + (a_n + b_n)x^2\Big) \\ &= (a_0 + b_0) x^2 + (a_1 + b_1) x^3 + \cdots + (a_n + b_n) x^{n + 2} \\ &= \Big(a_0 x^2 + a_1x^3 + \cdots + a_n x^{n + 2}\Big) + \Big(b_0 x^2 + b_1 x^3 + \cdots+ b_n x^{n + 2}\Big) \\ &= x^2 (a_0 + \cdots + a_n x^n) + x^2 (b_0 + \cdots + b_n x^n) \\ &= x^2 u + x^2 v \\ &= T(u) + T(v) \end{align*}

齐性同理可证