Show that if $V$ is a finite-dimensional vector space with a dot product $\langle-,-\rangle$, and $f: V \rightarrow V$ linear with $\forall v,w \in V: \langle v,w \rangle=0 \Rightarrow \langle f(v),f(w) \rangle=0$ then $\exists C \in \mathbb{R}$ such that $(C\cdot f)$ is a linear isometry.
Notes & Thoughts: $g$ is a linear isometry means $\forall v \in V: \lVert g(v)\rVert=v$
Visually the theorem makes sense, if orthogonal vectors remain orthogonal under $f$ then the angles remain, so the original vector just changes its length. (If I understand this correctly)