1. Prove the following theorem: Theorem 0.2 Any isometry of $\mathbb{R}^n $ is uniquely determined by the images of an affinely independent set of (n+1)-points, i.e., for an affinely independent set, an isometry $f$ is uniquely determined by $\{f(a_0),f(a_1),…,f(a_n) \}$.
sol) Since $f$ is an isometry between $\mathbf{R}^n$, there is an orthogonal matrix $A$ and a real vector $b$ such that $f(x) = Ax+b$.
Since an orthogonal matrix has a full rank, $\{A(a_i – a_0 )=f(a_i)-f(a_0) : i=1,2,..,n\}$ is an also basis of $\mathbb{R}^n$.
Therefore, for any $x \in \mathbb{R}^n$, there exist unique $c_i, i=1,…n$ such that
$f(x) = \sum_{i=1}^n c_i ( f(a_i)-f(a_0)=\sum_{i=1}^nc_i f(a_i)-\sum_{j=1}^n c_j f(a_0)$.
출처- 기하학1 (GIST-강현석교수님 강의) Homework 1 문제 중 일부