1.2.18 Determinant calculation

condition: Prove that if \( k_{1}, k_{2}, \ldots, k_{n-1}- \) are distinct natural numbers, \( a_{1}, a_{2}, \ldots, a_{n}- \) are distinct positive numbers, then the determinant \( \left|\begin{array}{cccc}1 & 1 & \ldots & 1 \\ a_{1}^{k_{1}} & a_{2}^{k_{1}} & \ldots & a_{n}^{k_{1}} \\ \ldots & \ldots & \ldots & \cdots \\ a_{1}^{k_{n-1}} & a_{2}^{k_{n-1}} & \ldots & a_{n}^{k_{n-1}}\end{array}\right| \) is non-zero.