= Reciprocal-basis coordinate transformation
{title2=$[v]_{W}=G_U[v]_{U}$}
In a real finite-dimensional <inner product space>, let two <bases> satisfy $\langle u_i,w_j\rangle=\delta_{ij}$. Taking <inner products> with $u_i$ in $v=\sum_jb_jw_j$ gives $b_i=\langle u_i,v\rangle$. If also $v=\sum_ja_ju_j$, then $b_i=\sum_j\langle u_i,u_j\rangle a_j$. Thus the <Gram matrix> $G_U$ converts the original <coordinates> into <reciprocal basis> coordinates, and $G_W=G_U^{-1}$ converts back. With Euclidean Cartesian column <basis matrices> $U,W$, reciprocity is $U^TW=I$, hence $W=U^{-T}$ and $W^{-1}U=U^TU=G_U$. This is why <dot products> with nonorthogonal <basis vectors> are coordinates in the reciprocal <basis>, rather than coordinates in the original <basis>.
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