General Approach to Computing Velocity of frames in Special Relativity?

Question

I've been self studying some special relativity by reading some textbooks, and I'm somewhat confused about the process of computing velocities. The book I'm reading primarily identifies computation of velocities by just carrying out the boost operators, but the general case remains unclear.
If you have a reference frame $pmatrix{x & t}$ at rest, and another reference frame $pmatrix{x' & t'}$ moving with respect to the other frame, and you have coordinate equations $x'(x,t)$ and $t'(x,t)$ relating the one frame to the other, how do you compute the relative velocity of the $pmatrix{x'&  t'}$ frame with respect to the $pmatrix{x & t}$ frame?
I'm struggling with some of the semantics, so maybe some elaboration on intuition might be appreciated too. Thanks.

robphy · Answer

Since for a boost
$$
left( 
begin{array}{c}
t'
x'
end{array}
right)
=
left( 
begin{array}{cc}
gamma & gamma v
gamma v  & gamma
end{array}
right)
left( 
begin{array}{c}
t
x
end{array}
right),
$$
given
$$
left( 
begin{array}{c}
t'
x'
end{array}
right)
=
left( 
begin{array}{cc}
A & B
B  & A
end{array}
right)
left( 
begin{array}{c}
t
x
end{array}
right),
$$
where $det(M)=A^2-B^2=1$,

the relative velocity is
$$frac{B}{A}=frac{gamma v}{gamma}=v.$$

General Approach to Computing Velocity of frames in Special Relativity?

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