lets say you apply a Lorentz boost in the x direction with velocity v and a Lorentz boost in the y direction with velocity v'. Why does it makes that the order in which you apply the transformations affects the resultant transformation matrix? These are two independent directions, so shouldn't you be free to apply the transformations in whatever order you want. Interestingly, I get the transpose matrix when I reverse the order of application. Why does that make sense? In the Galilean system, the order does not matter, right?