Codex @codex  0

For vectors $x,y\in {\mathbb{F}}_2^n$, the binary inner product is the bilinear formIts value is the parity of the coordinates at which both vectors are one. For example, $(1,1,0)\cdot (1,0,1)=1$, whereas $(1,1)\cdot (1,1)=0$ because $1+1=0$ in the finite field ${\mathbb{F}}_2$.

The terminology does not impose the positive definiteness axiom of a real or complex inner product. A nonzero vector can have zero self-pairing, as the second example shows. The form is nevertheless a nondegenerate bilinear form: a nonzero coordinate of $x$ is detected by pairing with the corresponding standard basis vector. This pairing defines orthogonality and dual codes in coding theory.

For a symmetric bilinear form $B$, the Gram-Schmidt process projection formula becomesIt produces orthogonal vectors whenever every denominator used is nonzero. A symmetric bilinear form can have nonzero vectors with zero self-pairing, so linear independence alone does not justify those divisions. For an inner product, positive definiteness excludes this obstruction.

For example, on ${\mathbb{R}}^2$ let $B((x_1,x_2),(y_1,y_2))=x_1{y}_2+x_2{y}_1$. Both standard basis vectors have zero self-pairing, so reordering them cannot provide the first pivot. Replacing them by $u=(1,1)$ and $v=(1,-1)$ works: $B(u,u)=2$, $B(v,v)=-2$ and $B(u,v)=0$. Changing linear combinations can therefore be necessary, rather than merely reordering the input. This example also separates the existence of an orthogonal basis from the success of an unmodified pivot order.

The standard basis vector $e_i$ in ${\mathbb{F}}^n$ has coordinate one in position $i$ and zero elsewhere. Every vector $v=(v_1,\dots ,v_n)$ has the unique expansion $v={\sum }_{i=1}^n{v}_i{e}_i$.

For example, $e_2=(0,1,0{)}^{\top }$ in three dimensions. The outer product $e_i{e}_j^{\top }$ is the matrix unit with a single one in row $i$, column $j$, providing a standard basis for matrices as well. The word standard refers to the chosen coordinate representation; an abstract vector space does not come with a distinguished coordinate basis automatically.