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Homework Statement:
 Find the matrix for the Lorentz transformation consisting of a boost of speed ##v## in the ##x##direction followed by a boost of speed ##w## in the ##y'## direction. Show that the boosts performed in the reverse order would give a different transformation.
Relevant Equations:

When it comes to Lorentz transformations, I feel like there are countless equations available for it. However, for this particular problem, I used the following equation for the Lorentz transformation and boost.
Lorentz Transformation:
$$S(P, P) = \eta_{\alpha \beta} x^{\alpha} x^{\beta} = x^{\alpha} \eta_{\alpha \beta} x^{\beta} = c^2 t^2 + x^2 + y^2 + z^2$$
Boost:
##ct' = Act + Bx## ##\Rightarrow## ##(ct') = (A, B, 0, 0) (ct)##
## x' = Cct + Dx## ##\Rightarrow## ##(x') = (C, D, 0, 0) (x)##
##y' = y## ##\Rightarrow## ##(y') = (0, 0, 1, 0) (y)##
##z' = z## ##\Rightarrow## ##(z') = (0, 0, 0, 1) (z)##
Where A is ultimately equal to 1, B = 0, C = 0, and D =1.
*Please note that the above equation for boost is supposed to be a matrix. However, typing it out on the forum didn't work very well for me. Perhaps there's a way to type out a matrix here that I don't know yet.
Unfortunately, I am not entirely confident of the above equations being able to do the trick and ultimately solve for the question. However, my guess is that using the equation written above for "boost", I could perhaps use ##v## and insert it into the ##x##direction part of the matrix (somehow) and do the same for the speed ##w## in the ##y'##direction which is on the other (left) side of the matrix equation for boost.
Furthermore, when it comes to Lorentz transformations, I believe the question would involve the Lorentz transformation identity ##I## during the calculation process since "something times its inverse is identity ##I##" (e.g. ##L \cdot L^{1} = I##).
Other than these small ideas, I have a difficult time knowing how to proceed with solving the question. Does anyone in the community perhaps have any helpful suggestions to help me move forward in solving the problem? Any help would be sincerely appreciated. Thank you very much for reading through this!
Furthermore, when it comes to Lorentz transformations, I believe the question would involve the Lorentz transformation identity ##I## during the calculation process since "something times its inverse is identity ##I##" (e.g. ##L \cdot L^{1} = I##).
Other than these small ideas, I have a difficult time knowing how to proceed with solving the question. Does anyone in the community perhaps have any helpful suggestions to help me move forward in solving the problem? Any help would be sincerely appreciated. Thank you very much for reading through this!