Let be $V$ a vectorial space and $B=\{ e_{1}, ..., e_{n} \}$ its basis and $B^{*} = \{ \phi_{1}, ..., \phi_{n}\}$ the dual basis of $B$. Determine the coordinates of a bilinear form $f \in \mathcal{B}(V\times V,K)$
If $V$ and $U$ are two vectorial spaces and $B = \{ v_{1},..., v_{n} \}$ and $C= \{u_{1},..., v_{m} \}$ are their basis, respectively, with $B^{*} = \{ h_{1}, ..., h_{n}\}$ and $C^{*} = \{ g_{1}, ..., g_{n}\}$ the dual basis of $B$ and $C$ a basis for $\mathcal{B}(V\times U,K)$ is $H= \{ f_{ij} = h_{i}g_{j} ; 1\leq i \leq n, 1\leq j \leq m \}$. In the question we have only the vectorial space $V$ so the basis for $\mathcal{B}(V;K)$ is $H= \{ f_{ij} = \phi_{i}\phi_{j} ; 1\leq i,j \leq n\}$. I'm with difficult to replace the vector of $V$ and find his coordinates. It is necessary to replace the vectors writting them with the dual basis?