Seriously struggling right now, and I could really use some help.
I am trying to follow this derivation of Lorentz Transformations from time dilation and length contraction below:
Time dilation and length contraction
The transformation equations can be derived from time dilation and length contraction, which in turn can be derived from first principles. With $O$ and $O'$ representing the spatial origins of the frames $F$ and $F'$, and some event $M$, the relation between the position vectors (which here reduce to oriented segments $OM$, $OO'$ and $O'M$) in both frames is given by: $$OM = OO' + O'M$$
Using coordinates $(x,t)$ in $F$ and $(x',t')$ in $F'$ for event $M$, in frame $F$ the segments are $OM = x$, $OO' = vt$ and $O'M = x'/\gamma$ (since $x'$ is $O'M$ as measured in $F'$): $$x=vt+x'/\gamma$$
Likewise, in frame $F'$, the segments are $OM = x/\gamma$ (since $x$ is $OM$ as measured in $F$), $OO' = vt'$ and $O'M = x'$: $$x/\gamma =vt'+x'$$
By rearranging the first equation, we get $$x'=\gamma (x-vt)$$ which is the space part of the Lorentz transformation. The second relation gives $$x=\gamma (x'+vt')$$ which is the inverse of the space part. Eliminating $x'$ between the two space part equations gives $$t'=\gamma (t-vx/c^{2})$$ which is the time part of the transformation, the inverse of which is found by a similar elimination of $x$: $$t=\gamma (t'+vx'/c^{2})$$
I cannot understand why $O'M = x'/\gamma$ or their reasoning of "since $x'$ is $O'M$ as measured in $F'$." I would seriously appreciate any help with this, as it has been driving me insane for more hours than I'd like to admit.