Let $k$ be a field, let $V$ and $W$ be finite-dimensional vector spaces over $k$, and let
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\begin{align*}
B: V \times W \to k
\end{align*}
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be a [bilinear form](/page/Bilinear%20Form). Let $(v_1,\dots,v_m)$ be a basis of $V$, let $(w_1,\dots,w_n)$ be a basis of $W$, and let $A \in k^{m \times n}$ be the matrix of $B$ in these bases, with entries
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\begin{align*}
A_{ij} = B(v_i,w_j).
\end{align*}
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If $\operatorname{rank}(B)$ denotes the rank of the associated [linear map](/page/Linear%20Map)
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\begin{align*}
\Psi_B: W \to V^*, \qquad \Psi_B(w)(v)=B(v,w),
\end{align*}