Suppose \(\mathscr{G}\) is the class of linear functions. A linear function evaluated at a feature \(\boldsymbol{x}\) can be described as \(g(\boldsymbol{x})=\boldsymbol{\beta}^{\top} \boldsymbol{x}\) for some parameter vector \(\beta\) of appropriate dimension. Denote \(g^{\mathscr{G}}(\boldsymbol{x})=\boldsymbol{x}^{\top} \boldsymbol{\beta}^{\mathscr{G}}\) and \(g_{\tau}^{\mathscr{G}}(\boldsymbol{x})=x^{\top} \widehat{\boldsymbol{\beta}}\). Show that

\[ \mathbb{E}\left(g_{\tau}^{\mathscr{G}}(\boldsymbol{X})-g^{*}(\boldsymbol{X})\right)^{2}=\mathbb{E}\left(\boldsymbol{X}^{\top} \widehat{\boldsymbol{\beta}}-\boldsymbol{X}^{\top} \boldsymbol{\beta}^{\mathscr{G}}\right)^{2}+\mathbb{E}\left(\boldsymbol{X}^{\top} \boldsymbol{\beta}^{\mathscr{G}}-g^{*}(\boldsymbol{X})\right)^{2} \]

Hence, deduce that the statistical error in (2.16) is \(\ell\left(g_{\tau}^{\mathscr{G}}\right)-\ell\left(g^{\mathscr{C}}\right)=\mathbb{E}\left(g_{\tau}^{\mathscr{G}}(\boldsymbol{X})-g^{\mathscr{G}}(\boldsymbol{X})\right)^{2}\).