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Inner product

Source: Iranian National Olympiad (3rd Round) 2006

September 21, 2006
linear algebralinear algebra unsolved

Problem Statement

Suppose (u,v)(u,v) is an inner product on Rn\mathbb R^{n} and f:Rn⟶Rnf: \mathbb R^{n}\longrightarrow\mathbb R^{n} is an isometry, that f(0)=0f(0)=0. 1) Prove that for each u,vu,v we have (u,v)=(f(u),f(v)(u,v)=(f(u),f(v) 2) Prove that ff is linear.
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