Let certain configuration of $n$ points exist in $d-$dimensional space, $X\in\mathbb{R}^{n\times d}$, $d<<n$. Also, let the corresponding Gram matrix be defined as $G=XX^T$.
Since $X$ exists in Euclidean space, rank of $G$ is, $rank(G)=rank(X)=d$. Now, suppose that the configuration $X$ is shifted to matrix $X'$ (shift corresponds to the origin translation). Could it be possible that the Gram matrix, $G'=X'(X')^T$, has rank different from $G$, ie, $rank(G)\neq rank(G')$?