14.9 Show that the random-coefficient model (14.2) is equi-variant under affine covariate transformation \(x \mapsto g_{0}+g_{1} x\) with \(g_{1} eq 0\). Show that the induced transformation on \(\left(\beta_{0}, \beta_{1}ight)\) is linear \(\mathbb{R}^{2} ightarrow \mathbb{R}^{2}\). Show that the induced transformation on variance components is also linear

\[
\left(\begin{array}{c}
\sigma_{0}^{2} \\
ho \sigma_{0} \sigma_{1} \\
\sigma_{1}^{2}
\end{array}ight) \longmapsto\left(\begin{array}{ccc}
1 & 2 g_{0} & g_{0}^{2} \\
0 & g_{1} & g_{0} g_{1} \\
0 & 0 & g_{1}^{2}
\end{array}ight)\left(\begin{array}{c}
\sigma_{0}^{2} \\
ho \sigma_{0} \sigma_{1} \\
\sigma_{1}^{2}
\end{array}ight)
\]