Let $Q: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be an operator that preserves all distances. Is this condition alone enough for us to say that $Q$ must be a linear operator?
If not, what are some counterexamples (whether simple or pathological), where an operator preserves all distances but is not a linear operator?
[Edit] Did I even give the correct interpretation of "preserves all distances"?
[Edit] I guess not! What I meant is that $Q$ is an isometry (i.e. $||Qx - Qy|| = ||x - y||$ for any $x$, $y$).