Prove, that f is a linear map.

Question

$U,V$ - Euclidean spaces
$f:U \rightarrow V$
$f(0)=0$
$ \forall _{u,v \in U}:d(f(u),f(v))=d(u,v)$

Prove that $f$ is a linear map.

I'm thinking about something like this: $||f(u+v)|| =d(f(u+v),0) = d(u+v,0) = ||u+v|| = ||u- (-v) || = d(u,-v) = d(f(u),f(-v)) = ||f(u) - f(-v)||$ but does it equeal $||f(u) + f(v)||$?

Answer 1

Hint Prove that

$$||f(\alpha u+v)-\alpha f(u)-f(v)||^2=0$$