设 $\alpha_i=\left(a_{i 1}, a_{i 2}, \cdots a_{i n}\right)^T(i=1,2, \cdots, r ; r < n)$ 是 $n$维实向量，且 $\alpha_1, \alpha_2, \cdots \alpha_r$ 线性无关. 己知 $\beta=\left(b_1, b_2, \cdots b_n\right)^T$是线性方程组
$$
\left\{\begin{array}{l}
a_{11} x_1+a_{12} x_2+\cdots a_{1 n} x_n=0 \\
a_{21} x_1+a_{22} x_2+\cdots a_{2 n} x_n=0 \\
\cdots \quad \cdots \cdots \cdots \cdots \\
a_{r 1} x_1+a_{r 2} x_2+\cdots a_{r n} x_n=0
\end{array}\right.
$$
的非零解向量. 试判断向量组 $\alpha_1, \alpha_2, \cdots \alpha_r, \beta$ 的线性相关性.