In my physics class, for deriving the Lorentz transformatuons my professor assumed that there is a linearity relatioin between the coordinates of the two systems. But why is that?? Can anyone please give a simple explanation? I couldn't find any.
They have to be linear transformations, otherwise the origin of the coordinate system would not be arbitrary as required by the homogeneity of space.
Another way to see this is to consider a particle moving along a straight line in one frame of reference. Under a non-linear transformation, this straight line would become curved, meaning that the particle is accelerated in that frame of reference which cannot be as the same laws are required to hold in all inertial systems by the principle of relativity.