I am reading Schwartz's "QFT and the standard model". On pg 13 he gives the Lorentz transform of a rotation around the x-axis:
$ \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos \theta _x & \sin \theta _x \\ 0 & 0 & -\sin \theta _x & \cos \theta _x \\ \end{array} \right) $
For a boost along the x-axis he gives:
$ \left( \begin{array}{cccc} \cosh \beta _x & \sinh \beta _x & 0 & 0 \\ \sinh \beta _x & \cosh \beta _x & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right) $
I believe this corresponds to the Lorentz transform in the form $\Lambda^\alpha{}_\beta$. I believe the first index is the row and the second is the column. But at the bottom of the pg he has $ V^\mu=\Lambda_\nu^\mu V^\nu. $ as though the order of the indices on $\Lambda$ doesn't matter. But surely they do matter as $\Lambda^\alpha{}_\beta\not=\Lambda_\beta{}^\alpha$ in general?