简单的例子

若 $\mathbf{A}=\left(\begin{array}{rr}1& 0\\ 1& -1\end{array}\right),\alpha =\left(\begin{array}{l}0\\ 1\end{array}\right),\beta =\left(\begin{array}{l}2\\ 1\end{array}\right),\gamma =\left(\begin{array}{l}1\\ 1\end{array}\right)$, 则有

$$\mathbf{A}\alpha =\left(\begin{array}{rr}1& 0\\ 1& -1\end{array}\right)\left(\begin{array}{l}0\\ 1\end{array}\right)=\left(\begin{array}{r}0\\ -1\end{array}\right)=-\alpha$$ $$\mathbf{A}\beta =\left(\begin{array}{rr}1& 0\\ 1& -1\end{array}\right)\left(\begin{array}{l}2\\ 1\end{array}\right)=\left(\begin{array}{l}2\\ 1\end{array}\right)=\beta$$

$\mathbf{A}\gamma =\left(\begin{array}{rr}1& 0\\ 1& -1\end{array}\right)\left(\begin{array}{l}1\\ 1\end{array}\right)=\left(\begin{array}{l}1\\ 0\end{array}\right)\ne k\gamma ,k\in \mathbb{R}$