设 $\alpha_1=\left(\begin{array}{c}a_1+b \\ a_1 \\ \vdots \\ a_1\end{array}\right), \alpha_2=\left(\begin{array}{c}a_2 \\ a_2+b \\ \vdots \\ a_2\end{array}\right), \cdots, \alpha_n=\left(\begin{array}{c}a_n \\ a_n \\ \vdots \\ a_n+b\end{array}\right)$. 记 $W=L\left(\alpha_1, \alpha_2, \cdots, \alpha_n\right)$ ，其中 $\sum_{i=1}^n a_i \neq 0$ ，求 $W$ 的维数与一组基.