A bilinear form may look like a scalar-valued function of two variables, but its useful structure is linear. If one input is fixed, the other input becomes a linear functional. The rank of the bilinear form measures how many independent functionals are produced in this way. It separates the directions detected by the pairing from the directions that pair to zero against everything.

This invariant connects [Bilinear Form](/page/Bilinear%20Form), [Vector Space](/page/Vector%20Space), [Linear Map](/page/Linear%20Map), and [Dual Space](/page/Dual%20Space). In coordinates it is ordinary matrix rank, but its meaning is coordinate-free: it measures the effective dimension of the pairing. It is the basic numerical invariant behind radicals, nondegeneracy, symmetric forms, alternating forms, and quadratic forms.

## Definition

The page is about a numerical invariant, so the first definition isolates that invariant directly. Fixing the left input of a bilinear form produces a linear functional on the right-hand space; the rank counts the dimension of the space of functionals obtained in this way.

[definition: Rank of a Bilinear Form] Let $V$ and $W$ be finite-dimensional vector spaces over a field $k$, and let $B: V \times W \to k$ be a bilinear form. The rank of $B$ is \begin{align*} \operatorname{rank}(B) := \dim \{\lambda \in W^* : \text{there exists } v \in V \text{ such that } \lambda(w)=B(v,w) \text{ for all } w \in W\}. \end{align*} [/definition]

The standard dot product shows the intended meaning of the definition before any machinery is introduced. Every coordinate direction produces an independent test functional, so the rank reaches the full ambient dimension. Since the associated maps and radicals have not yet been named, the example describes zero-pairing directions directly in terms of vectors that pair to zero against every vector on the opposite side.

[example: Standard Dot Product Has Full Rank] Let $k$ be a field and let $V=k^n$. Define $B:k^n \times k^n \to k$ by \begin{align*} B(x,y)=\sum_{i=1}^n x_i y_i. \end{align*} For the standard basis vectors $e_1,\ldots,e_n$, if $i=j$ then $B(e_i,e_j)=1$, and if $i \neq j$ then $B(e_i,e_j)=0$. Thus the matrix of $B$ in the standard basis is $I_n$. For each $i$, the functional on $k^n$ produced by fixing $e_i$ is \begin{align*} \lambda_i(y)=B(e_i,y)=y_i. \end{align*} Therefore $\lambda_i$ is the $i$th coordinate functional on $k^n$. These $n$ coordinate functionals form the standard basis of the dual space $(k^n)^*$, so the functionals obtained by fixing the first input span all of $(k^n)^*$. Hence \begin{align*} \operatorname{rank}(B)=\dim (k^n)^*=n. \end{align*} Now suppose a vector $x \in k^n$ pairs to zero against every vector $y \in k^n$. Taking $y=e_i$ gives \begin{align*} 0=B(x,e_i)=x_i. \end{align*} This holds for every $i$, so $x=0$. The same argument on the right gives: if $y \in k^n$ pairs to zero against every vector $x \in k^n$, then taking $x=e_i$ gives \begin{align*} 0=B(e_i,y)=y_i. \end{align*} Thus $y=0$. The standard dot product detects every nonzero vector on both sides; later definitions will call this two-sided detection nondegeneracy. [/example]

## Background and Associated Maps

The rank definition assumes that the two-variable pairing can be probed by fixing one input and reading off a linear functional in the other. Bilinearity is the precise condition that guarantees this probe is legitimate. Without it, the values of a two-variable function need not assemble into linear maps into dual spaces, so the matrix-style notion of rank would not survive a change of coordinates.

[definition: Bilinear Form] Let $k$ be a field, and let $V$ and $W$ be vector spaces over $k$. A bilinear form on $V \times W$ is a function $B: V \times W \to k$ such that for every $v, v_1, v_2 \in V$, every $w, w_1, w_2 \in W$, and every $a,b \in k$, \begin{align*} B(av_1+bv_2,w)=aB(v_1,w)+bB(v_2,w) \end{align*} and \begin{align*} B(v,aw_1+bw_2)=aB(v,w_1)+bB(v,w_2). \end{align*} [/definition]

The values of $B$ are scalars, so they cannot be counted directly as vectors in $V$ or $W$. The usable object is obtained by holding the left input fixed: each $v \in V$ then determines a linear functional on $W$, and the collection of all such functionals records exactly which tests on $W$ are produced by vectors of $V$.

To make rank a property of the pairing rather than a loose collection of scalar evaluations, this assignment has to be treated as an actual linear map. Naming that map gives a precise object whose image measures how many independent tests on $W$ come from the left input.

[definition: Left Linear Map of a Bilinear Form] Let $k$ be a field, let $V$ and $W$ be vector spaces over $k$, and let $B: V \times W \to k$ be a bilinear form. The left linear map associated to $B$ is the linear map $L_B: V \to W^*$ defined by \begin{align*} L_B(v)(w)=B(v,w) \end{align*} for all $v \in V$ and $w \in W$. [/definition]

A complete rank theory must not privilege the first input without checking the second. Rectangular pairings may detect $V$ and $W$ in different ways, so we need the companion map from $W$ into $V^*$. This map records which functionals on $V$ arise from vectors in $W$.

[definition: Right Linear Map of a Bilinear Form] Let $k$ be a field, let $V$ and $W$ be vector spaces over $k$, and let $B: V \times W \to k$ be a bilinear form. The right linear map associated to $B$ is the linear map $R_B: W \to V^*$ defined by \begin{align*} R_B(w)(v)=B(v,w) \end{align*} for all $v \in V$ and $w \in W$. [/definition]

Rank is meaningful because some directions may be invisible to all tests supplied by the opposite space. We need a name for the invisible vectors on the left side. This subspace will later become the kernel term in the rank-nullity formula for $B$.

[definition: Left Radical of a Bilinear Form] Let $k$ be a field, let $V$ and $W$ be vector spaces over $k$, and let $B: V \times W \to k$ be a bilinear form. The left radical of $B$ is the subspace \begin{align*} \operatorname{rad}_L(B) := \{v \in V : B(v,w)=0 \text{ for all } w \in W\}. \end{align*} [/definition]

The right input can have its own invisible vectors. This separate definition is necessary because a form can be nondegenerate on one side and degenerate on the other when the spaces have different dimensions.

[definition: Right Radical of a Bilinear Form] Let $k$ be a field, let $V$ and $W$ be vector spaces over $k$, and let $B: V \times W \to k$ be a bilinear form. The right radical of $B$ is the subspace \begin{align*} \operatorname{rad}_R(B) := \{w \in W : B(v,w)=0 \text{ for all } v \in V\}. \end{align*} [/definition]

The initial definition of rank can now be rewritten using the associated left map: