Show that the transformation \(\mathbb{R}^{n} ightarrow \mathbb{R}^{n}\) defined by

\[
\bar{u} \mapsto \bar{u}+\text { const }, \quad u_{i}-\bar{u} \mapsto \lambda\left(u_{i}-\bar{u}ight)
\]

is linear and invertible with Jacobian \(J=|\lambda|^{n-1}\). Here, \(\bar{u}\) is the mean of the components of the vector \(u \in \mathbb{R}^{n}\), and \(\lambda\) is a non-zero constant.