Source for if $L_A:(\mathbb{R}^n, \left\Vert \cdot \right\Vert_p) \to (\mathbb{R}^n, \left\Vert \cdot \right\Vert_q)$ is an isometry, then $p = q$.
$(\mathbb{R}^n,\|\cdot\|_{p})$ is isometrically isomorphic to $(\mathbb{R}^n,\|\cdot\|_{q})$ iff $p=q,$