Gramian Matrix - Gram Determinant

The Gram determinant or Gramian is the determinant of the Gram matrix:

$G(x_1,\dots, x_n)=\begin{vmatrix} \langle x_1,x_1\rangle & \langle x_1,x_2\rangle &\dots & \langle x_1,x_n\rangle\\ \langle x_2,x_1\rangle & \langle x_2,x_2\rangle &\dots & \langle x_2,x_n\rangle\\ \vdots&\vdots&\ddots&\vdots\\ \langle x_n,x_1\rangle & \langle x_n,x_2\rangle &\dots & \langle x_n,x_n\rangle\end{vmatrix}.$

Geometrically, the Gram determinant is the square of the volume of the parallelotope formed by the vectors. In particular, the vectors are linearly independent if and only if the Gram determinant is nonzero (if and only if the Gram matrix is nonsingular).

The Gram determinant can also be expressed in terms of the exterior product of vectors by

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