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

Any comments on this question are most appreciated!

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!