Let $R$ be a Noetherian regular local ring of dimension $n$ with maximal ideal $\mathfrak{m}$. Given two systems of regular parameters $\vec{u}=\left<u_1, \dots, u_n\right>$ and $\vec{v}=\left<v_1, \dots, v_n\right>$ of $R$, does there exist an element $M$ in $\mathrm{GL}_n(R)$ which takes $\vec{u}$ to $\vec{v}$? 

Any comments on this question are most appreciated!