The question comes from Kaplansky's book Linear Algebra and Geometry on page 96 exercise 2
Let $V$ be a non-singular inner product space of characteristic $\neq2$. Let $T$ be a one-to-one map of $V$ onto itself, sending $0$ to $0$ and satisfying $(x-y, x-y) = (Tx - Ty, Tx - Ty)$ for all $x,y \in V$. Prove that $T$ is orthogonal.
That $T$ preserves inner products is easy, but I don't know how to prove that it is linear. Any help would be appreciated.