Solution
= Solution
The simple reflections act on the root basis by
$$
\begin{array}{c|ccc}
&\alpha_1&\alpha_2&\alpha_3\\ \hline
s_{\alpha_1}&-\alpha_1&\alpha_2+\alpha_1&\alpha_3+\alpha_1\\
s_{\alpha_2}&\alpha_1+\alpha_2&-\alpha_2&\alpha_3\\
s_{\alpha_3}&\alpha_1+\alpha_3&\alpha_2&-\alpha_3
\end{array}
$$
by the <Weyl reflection> formula $s_\alpha(\beta)=\beta-\langle\beta,\alpha^\vee\rangle\alpha$.