Isometry in R^3/4 axes to themselves/No 3 in plane/Identity/Exercise

Let ${}A_{1},A_{2},A_{3}$ and ${}A_{4}$ be four lines in ${}\mathbb {R} ^{3}$ through the origin fulfilling the property that no three of them lie in a plane. Let

$f\colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3}$

be a linear, proper isometry with ${}f(A_{i})=A_{i}$ for ${}i=1,2,3,4$. Show that ${}f$ is the identity. Give an example to show that this statement is not true without the plane condition.

Create a solution