If $B$ and $B'$ are the matrix representations of a bilinear form in two bases, then these matrices are related by the equation $T^t B T = B'$ for an invertible matrix $T$.
Is it the case that automatically $T^t = T^{- 1}$ for bilinear forms, which yields the well-known transformation of matrix representations of endomorphisms under a change in basis, or does the basis transformation for bilinear forms only work for $T^t B T = B'$ and not necessarily for $T^{-1}BT=B'$?