Show that every isometry of $\mathbb{R}^n$ is an affine map.
sol) $f$ has a form $f(x) = Ax +b $. Let $x_1,x_2 \in \mathbb{R}^n$ and $c_1, c_2 \in \mathbb{R}$ such that $c_1+c_2=1$. $f(c_1x_1+c_2x_2)=A(c_1x_1+c_2x_2)+b= A(c_1x_1)+A(c_2x_2)+(c_1+c_2)b=c_1Ax_1+c_1b+c_2Ax_2+c_2b=c_1f(x_1)+c_2f(x_2)$.
출처- 기하학1 (GIST-강현석교수님 강의) Homework 1 문제 중 일부