Solution
= Solution
The <Weyl reflection> formula $s_i(\beta)=\beta-\langle\beta,\alpha_i^\vee\rangle\alpha_i$, together with the <Cartan matrix>, gives
$$
\begin{array}{c|ccc}
&\alpha_1&\alpha_2&\alpha_3\\ \hline
s_1&-\alpha_1&\alpha_1+\alpha_2&\alpha_3\\
s_2&\alpha_1+\alpha_2&-\alpha_2&2\alpha_2+\alpha_3\\
s_3&\alpha_1&\alpha_2+\alpha_3&-\alpha_3
\end{array}.
$$
These are \b[the images of every simple root under each simple reflection].