# Deriving Linear Transformations - Special Relativity

I wasn't sure if this counted as intro physics. Feel free to move if I have it in the wrong place.

## Homework Statement

In class we learned some linear transformations where we have a stationary observer and another moving near the speed of light.

Describing the reference frames:
s' -> x'=u't'
s-> x=ut

Boundary conditions:
x=x'=t=t'=0

Therefore at the origin of x we have x=ut=0, x'=u't', x=-vt'

Using these for the transformations:
x'=Ax+Bt
t'=Cx+Dt

## Homework Equations

Final transforms:

$$x'=\gamma(x-vt)$$

$$t'=\gamma(\frac{-vx}{c^2}+t)$$

$$x=\gamma(x'+vt')$$

$$t=\gamma(\frac{vx'}{c^2}+t')$$

## The Attempt at a Solution

So I can solve these as long as I have my notes that I can follow so I know what to solve for and when. My professor ended up solving for the A,B,C,D first then subbing in gamma etc to arrive at the final results.

My questions is: I'd love to be able to derive these on my own. My problem I don't know what I am trying to solve for or the point of what I am solving for. Point may not be the best word - I understand the end result and know how it is used, I just don't understand the steps/method to get there. (the algebra I get, not the method.