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Thomas
Thomas
20:20
The purpose of the whole exercise is to determine the coefficients of the proposed coordinate transformation

$$x'=Dx+Kct$$
$$ct'=Act+Bx$$

You can not make any a priori assumptions about the coefficients but you have to derive them by applying the appropriate constraints. Usually, the constraint is taken as the squared form of the light path equation

$$x^2=c^2t^2 ;x'^2=c^2t'^2$$

which results in the coefficients we are familiar with from the Lorentz transformation (as detailed in my answer).
 
1 hour later…
J. Murray
J. Murray
21:50
@Thomas Okay. I have attempted to explain several times what the problem is. When you write $x_2=-x_1$, you are implicitly evaluating the positions of the light beams at the same time. When you boost to a new frame, those two events are no longer simultaneous, which is why $x_2'\neq -x_1'$.
I strongly suggest you ask a separate question about this. You have a deep, fundamental misunderstanding of Lorentz transformations and special relativity, and this misunderstanding is going to severely limit your ability to understand modern physics. I'm sorry this conversation hasn't been sufficient to fix this misunderstanding, but I hope (a) that you're open to trying to understand it, and (b) that someone else can do a better job.

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