Here is a decomposition of galilean transforms of the form $x\mapsto Ax+y.$ Why are they all of this form?
$T$ galilean is distance preserving so it is also injective. Take $B_r(a)$ the closed $r-$ball at $a$, similarly $B_s(b)$ with $r+s=|a-b|.$ Then just as $B_r(a)\cap B_s(b)$ is singleton, their inverse-maps under $T$ must also be singleton. Varying $r,s$ holding their sum constant, one sees the inverse map of the line segment between $a$ and $b$ is another line segment. So $T$ preserves lines. But why must $T$ have the above formula?
