Suppose $(u,v)$ is an inner product on $\mathbb R^{n}$ and $f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is an isometry, that $f(0)=0$. 1) Prove that for each $u,v$ we have $(u,v)=(f(u),f(v)$ 2) Prove that $f$ is linear.
Problem
Source: Iranian National Olympiad (3rd Round) 2006
Tags: linear algebra, linear algebra unsolved