Let $T : \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be an orthogonal transformation such that $\operatorname{det} T = 1$ and $T$ is not the identity linear transformation. Let $S \subset \mathbb{R}^3$ be the unit sphere, i.e.,
$$S = \left\{ (x, y, z) \mid x^2 + y^2 + z^2 = 1 \right\}$$
Show that $T$ fixes exactly two points on $S$.