  In the derivation of Lorentz transformation, linear transformation between
inertial frames is one of the most important steps. In teaching special
relativity, we usually use the homogeneity and isotropy of spacetime to argue
that the transformation must be linear transformation without providing any
rigorous detail. Here in the first time we provide a solid mathematical proof
of the argument that the transformation between two inertial frames must be
linear because of the homogeneity and isotropy of spacetime.
