matrix representation of a bilinear form

Given a bilinear form, $B:U\times V\to K$, we show how we can represent it with a matrix, with respect to a particular pair of bases for $U$ and $V$

Suppose $U$ and $V$ are finite-dimensional and we have chosen bases, ${ℬ}_1=\{e_1,\mathrm{\dots }\}$ and ${ℬ}_2=\{f_1,\mathrm{\dots }\}$. Now we define the matrix $C$ with entries $C_{ij}=B(e_i,f_j)$. This will be the matrix associated to $B$ with respect to this basis as follows; If we write $x,y\in V$ as column vectors in terms of the chosen bases, then check $B(x,y)=x^{T}Cy$. Further if we choose the corresponding dual bases for $U^{\ast }$ and $V^{\ast }$ then $C$ and $C^T$ are the corresponding matrices for $B_R$ and $B_L$, respectively (in the sense of linear maps). Thus we see that a symmetric bilinear form is represented by a symmetric matrix, and similarly for skew-symmetric forms.

Let ${ℬ}_1^{\prime }$ and ${ℬ}_2^{\prime }$ be new bases, and $P$ and $Q$ the corresponding change of basis matrices. Then the new matrix is $C^{\prime }=P^{T}CQ$.