behaviour of E and B under Lorentz transformations

In a frame $\mathcal{O}'\,$ moving in the direction $\hat\mathbf{v}\,$ ,

$\mathbf{E}_\parallel' = \mathbf{E}_\parallel\,$					$\mathbf{E}_\perp' = \gamma(\mathbf{E}_\perp + \mathbf{v} \times\mathbf{B}_\perp )\,$
$\mathbf{B}_\parallel' = \mathbf{B}_\parallel\,$					$\mathbf{B}_\perp' = \gamma(\mathbf{B}_\perp - \mathbf{v}\times\mathbf{E}_\perp )\,$