I have a question about the derivation of Lorentz transformations. My syllabus starts the proof in the following way: We have two inertial reference frames, S and S', that coincide at their origins at t=t'=0. S' is moving in the positive x-direction relative to S with velocity $v$. At t=0, an observer at the origin of S shines a light beam in the positive x-direction. We obtain $x-ct=x'-ct'$. This is still true if we consider a more general form, $x-ct=\lambda(x'-ct')$, where $\lambda$ is yet to be determined. So far, so good. Now they say that this observer simultaneously also shines a light beam in the negative x-direction, which yields to $x+ct=x'+ct'$. However, the syllabus now says that we would get a different general form $x+ct=\mu (x'+ct')$, instead of using the same $\lambda$ again.
How is this possible? If we found $\lambda$, then it should make sense that that same $\lambda$ would work in the negative direction. Could someone explain to me why we're working with two unknowns here?
In case it helps, this is the proof as written in the syllabus (it's in Dutch). It's basically the "From physical principles" derivation found at the wikipedia page: https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations