 Solution

The official notes state only ${\mathcal{F}}^{\prime }=\Lambda \mathcal{F}{\Lambda }^T$ (PDF p. 15, handwritten p. 10); they do not evaluate the requested coefficients. For a boost along positive $x$, use ${\Lambda }^0{}_0={\Lambda }^1{}_1=\gamma$, ${\Lambda }^0{}_1={\Lambda }^1{}_0=-\gamma v/c$, with the transverse diagonal entries equal to one. Multiplying the specified electromagnetic field tensor givesThe corresponding Lorentz transformation of electromagnetic fields isConsequentlyThese coefficients depend on the frame speed, but are constant for the fixed boost.