已知 $R^3$ 的两个基为
$$
\alpha_1=\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right], \alpha_2=\left[\begin{array}{l}
1 \\
0 \\
-1
\end{array}\right], \alpha_3=\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right]
$$

$$
\beta_1=\left[\begin{array}{l}
1 \\
2 \\
1
\end{array}\right], \beta_2=\left[\begin{array}{l}
2 \\
3 \\
4
\end{array}\right], \beta_3=\left[\begin{array}{l}
3 \\
4 \\
3
\end{array}\right]
$$

求由基 $\alpha_1, \alpha_2, \alpha_3$ 到基 $\beta_1, \beta_2, \beta_3$ 的过渡矩阵.