Solution

To solve this problem we need to use the Lorentz transformation for fields, taking into account that the fields of the electromagnetic wave are
perpendicular to the velocity of the moving frame. Also, in each frame, the relation $\vec{E} = [\vec{v} \times \vec{B}]$ holds, where $\vec{v}$ is the
velocity vector of the electromagnetic wave; in this case $\vec{v} = \vec{c}$.

\begin{equation} E' = \frac{E - \beta c B}{\sqrt{1-\beta^2}} = E \sqrt{\frac{1+\beta}{1-\beta}} \end{equation}
We have take into account $[\vec{c} \times \vec{B}]$
is antiparallel with $\vec{E}$. And using the relation between fields:

\begin{equation} B' = B \sqrt{\frac{1+\beta}{1-\beta}} \end{equation}

Thus, the amplitude becomes $\sqrt{\frac{1-\beta}{1+\beta}}$ times smaller.

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