Let $\mathbf{v}_{1},\mathbf{v}_{2},\cdots,\mathbf{v}_{m}$ be $m$ vectors in $n$-dimensional space. Their Gram determinant is defined by:
$$\Gamma=\left|\begin{array}{cccc} \mathbf{v}_{1}^{2} & \left(\mathbf{v}_{1}\cdot\mathbf{v}_{2}\right) & \cdots & \left(\mathbf{v}_{1}\cdot\mathbf{v}_{m}\right)\\ \left(\mathbf{v}_{2}\cdot\mathbf{v}_{1}\right) & \mathbf{v}_{2}^{2} & \cdots & \left(\mathbf{v}_{2}\cdot\mathbf{v}_{m}\right)\\ \cdots & \cdots & \cdots & \cdots\\ \left(\mathbf{v}_{m}\cdot\mathbf{v}_{1}\right) & \left(\mathbf{v}_{m}\cdot\mathbf{v}_{2}\right) & \cdots & \mathbf{v}_{m}^{2} \end{array}\right|$$
If $v_{ij}$ is $j$th component of $\mathbf{v}_{i}$, prove that
$$\Gamma=\sum\left|\begin{array}{cccc} v_{1s_{1}} & v_{1s_{2}} & \cdots & v_{1s_{m}}\\ v_{2s_{1}} & v_{2s_{2}} & \cdots & v_{2s_{m}}\\ \cdots & \cdots & \cdots & \cdots\\ v_{ms_{1}} & v_{ms_{2}} & \cdots & v_{ms_{m}} \end{array}\right|^{2}$$
where the summation is extended over all integers $s_{1},s_{2},\cdots,s_{m}$ from 1 to $n$ with $s_{1}