设 $t_1,t_2,\dots,t_r$是互不相同的数,设 $\alpha_i=(1,t_i,{t_i}^2,\dots,{t_i}^{n-1})(i=1,2,\dots,r)$
讨论向量组
$\alpha_1,\alpha_2,\dots,\alpha_r$
的线性相关性.

解：(1)当 $r>n$时, $r$个 $n$维向量必线性相关;
(2)当 $r\leq n$时,将 $\alpha_1,\alpha_2,\dots,\alpha_r$按行排列成矩阵
$A=\left(\begin{matrix} \alpha_1\\ \alpha_2\\ \vdots\\ \alpha_r\\ \end{matrix}\right)=\left(\begin{matrix} 1&t_1&{t_1}^2&\dots&{t_1}^{n-1}\\ 1&t_2&{t_2}^2&\dots&{t_2}^{n-1}\\ \vdots&\vdots&\vdots& &\vdots\\ 1&t_r&{t_r}^2&\dots&{t_r}^{n-1}\\ \end{matrix}\right)$
$A$中 $r$阶子式(范德蒙行列式的转置)
$D_r=\left(\begin{matrix} 1&t_1&{t_1}^2&\dots&{t_1}^{r-1}\\ 1&t_2&{t_2}^2&\dots&{t_2}^{r-1}\\ \vdots&\vdots&\vdots& &\vdots\\ 1&t_r&{t_r}^2&\dots&{t_r}^{r-1}\\ \end{matrix}\right)=\prod_{r\ge i\ge j\ge 1}(t_i-t_j)\not = 0$
因 $r(A)=r,$故
$\alpha_1,\alpha_2,\dots,\alpha_r线性无关.$