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If $G$ is the Gram matrix of $n$ vectors of size $d$ that are linearly independent ($n\leq d$), then $G$ is invertible and its inverse is a Gram matrix. Of what?

To be more precise, if $G$ is the Gram matrix of $u_1,\ldots,u_n$, where each $u_i\in\mathbb R^d$ and $u_1,\ldots,u_n$ are linearly independent, then there exists a collection of $n$ linearly independent vectors $v_1,\ldots,v_n$ whose $G^{-1}$ is the Gram matrix. Is there some relationship between the $u$'s and the $v$'s?

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yes, $$ \langle u_i, v_j \rangle = \delta_{ij} $$ meaning $1$ if $i=j$ but $0$ otherwise

The convention in Riemannian geometry is to write the elements of the original $G$ as $g_{ij},$ and the elements of the inverse matrix as $g^{ij}.$ The original basis is often $e_i,$ while the new basis is $f^j.$ The construction is $$ f^i = \sum_j g^{ij} e_j $$

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