Let $k$ be a field, let $V$ be a [vector space](/page/Vector%20Space) over $k$, let $W$ be a finite-dimensional vector space over $k$, and let $B: V \times W \to k$ be a $k$-[bilinear form](/page/Bilinear%20Form). Define the associated $k$-[linear map](/page/Linear%20Map) $\Phi_B: V \to W^*$ by $\Phi_B(v)(w) = B(v,w)$ for all $v \in V$ and $w \in W$, and define $\operatorname{rank}(B) := \dim_k \operatorname{im}(\Phi_B)$. Then $\operatorname{rank}(B)$ is the least integer $r \geq 0$ such that there exist a finite-dimensional $k$-vector space $U$ with $\dim_k U = r$, a $k$-linear map $a: V \to U$, and a $k$-bilinear map $C: U \times W \to k$ satisfying

\begin{align*} B(v,w) = C(a(v),w) \end{align*}

for all $v \in V$ and $w \in W$.