已知 $\mathbf{R}^3$ 的两个基为 $\boldsymbol{\alpha}_1=\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right), \boldsymbol{\alpha}_2=\left(\begin{array}{c}1 \\ 0 \\ -1\end{array}\right), \boldsymbol{\alpha}_3=\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right)$ 与 $\boldsymbol{\beta}_1=\left(\begin{array}{l}1 \\ 2 \\ 1\end{array}\right), \boldsymbol{\beta}_2=\left(\begin{array}{l}2 \\ 3 \\ 4\end{array}\right)$ ， $\boldsymbol{\beta}_3=\left(\begin{array}{l}3 \\ 4 \\ 3\end{array}\right)$ ．求由基 $\boldsymbol{\alpha}_1, \boldsymbol{\alpha}_2, \boldsymbol{\alpha}_3$ 到基 $\boldsymbol{\beta}_1, \boldsymbol{\beta}_2, \boldsymbol{\beta}_3$ 的过渡矩阵 $\boldsymbol{P}$ ．