= Solution
Use the $m$-dimensional space $V_m$ for this <change of basis>. With the columns of $P$ recording the new basis in the old one, bilinearity gives
$$
\begin{aligned}
\varphi(f_i,f_j)
&=\varphi\left(\sum_{k=1}^mP_{ki}e_k,\sum_{\ell=1}^mP_{\ell j}e_\ell\right)\\
&=\sum_{k,\ell=1}^mP_{ki}(A_m)_{k\ell}P_{\ell j}\\
&=(P^TA_mP)_{ij}.
\end{aligned}
$$
Thus the matrices of the <bilinear form> satisfy
$$
\boxed{B_m=P^TA_mP.}
$$
This is <matrix congruence>, rather than the similarity rule for an endomorphism: both arguments of the bilinear form change basis.
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