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