Let $k$ be a field, let $V$ and $W$ be finite-dimensional vector spaces over $k$, and let $B: V \times W \to k$ be a [bilinear form](/page/Bilinear%20Form). Let $\operatorname{rank}(B)$ denote the common rank of the associated linear maps $L_B: V \to W^*$, defined by $L_B(v)(w) = B(v,w)$, and $R_B: W \to V^*$, defined by $R_B(w)(v) = B(v,w)$.

Then $B$ is left nondegenerate if and only if

\begin{align*} \operatorname{rank}(B)=\dim V. \end{align*}

The form $B$ is right nondegenerate if and only if

\begin{align*} \operatorname{rank}(B)=\dim W. \end{align*}

Consequently, $B$ is nondegenerate, meaning both left and right nondegenerate, if and only if $\dim V=\dim W$ and

\begin{align*} \operatorname{rank}(B)=\dim V=\dim W. \end{align*}