This seems like an intuitive explanation to me posted by @Dale. This is quoted from this thread, post 24:Now, we are particularly interested in inertial coordinate systems, that is we like coordinates where free particles go in a straight line at constant velocity, following Newton's first law. Any coordinates where all free particles have straight lines as their worldlines are inertial coordinates, so if we want to study the transformations from one inertial frame to another inertial frame then we want to study transformations that map straight lines to other straight lines.
The simplest such transformation is a linear transformation, which is the form chosen in the derivation you cited. So the reason for choosing that generalization is that it is the simplest generalization that has the necessary property of mapping straight lines to straight lines.
Well, first semi-argument (weak) is: moving to the limit v/c -> 0, one should recover the space-time linear Galilei transformations. Next semi-argument (a stronger one): assume they are quadratic, that isThere are several ways to show that the Lorentz transformations must be linear. What's the best/more intuitive argument in your opinion?
It's not true to say that they "must" be linear.There are several ways to show that the Lorentz transformations must be linear. [...]