Let $\displaystyle T: \mathbb{R}^{3} \to \mathbb{R}^{3}$ be a linear transformation such that $\displaystyle det \ T=1$. If T is not the identity linear transformation and $\displaystyle T:S \to \mathbb{R}^{3}$ where $\displaystyle S=\{ (x,y,z) \in \mathbb{R}^{3} | x^{2}+y^{2}+z^{2}=1\}$.Then prove that T fixes exactly 2 points of S.