The standard bilinear form $⟨x,y⟩={\sum }_{i<n}{x}_i{y}_i$ on $F^n$

Fix a field $F$ and a natural number $n$. The standard bilinear form on $F^n$ is

$⟨x,y⟩:={\sum }_{i<n}{x}_i{y}_i\mathrm{.}$

The symbol denotes the finite sum in the additive commutative group of the field $F$: it is $0_F$ at $n=0$ and is obtained by successively adding the terms $x_i{y}_i$ for $i<n$.

This is a symmetric bilinear form in the sense of Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms, and its matrix in the standard basis of $F^n$ is the identity matrix. Hence it is nondegenerate in the sense of The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space.

When $F={\mathbb{F}}_2$, the squaring law $t^2=t$ gives

$⟨x,x⟩={\sum }_{i<n}{x}_i\mathrm{.}$

So over ${\mathbb{F}}_2$ the form detects parity and not positivity. It is not an inner product there, and the page never treats it as one.

• The formula is the same over every field, but the consequences are not. The Oddtown and Eventown arguments use only bilinearity over ${\mathbb{F}}_2$; the Fisher argument later uses the order and positivity of $\mathbb{R}$ as well.