Maybe this is a simple question. But I'm failed to see this.
Let $f:\mathbb{R}^{p}\to\mathbb{R}$, $$ f\left(\boldsymbol{\beta}\right)=\frac{1}{2n}\left\Vert \mathbf{e}-\mathbf{X}\left(\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\right)\right\Vert _{2}^{2}+\frac{1}{2}\left\Vert \boldsymbol{\beta}-\boldsymbol{\beta}_{1}\right\Vert _{2}^{2} $$ with the minimizer $\hat{\boldsymbol{\beta}}$. Now I transform the function by converting $\boldsymbol{\beta}$ terms to $\boldsymbol{\alpha}=\sqrt{n}\left(\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\right)$ to get $$ g\left(\boldsymbol{\alpha}\right)=\frac{1}{2n}\left\Vert \mathbf{e}-\frac{1}{\sqrt{n}}\mathbf{X}\boldsymbol{\alpha}\right\Vert _{2}^{2}+\frac{1}{2}\left\Vert \frac{\boldsymbol{\alpha}}{\sqrt{n}}+\boldsymbol{\beta}_{0}-\boldsymbol{\beta}_{1}\right\Vert _{2}^{2} $$ Is $\hat{\boldsymbol{\alpha}}=\sqrt{n}\left(\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}_{0}\right)$ the minimizer of $g\left(\boldsymbol{\alpha}\right)$ function?