= Bilinear form
{title2=$B:V\times V\to F$}
{wiki}
A <bilinear form> takes two <vectors> in a <vector space> $V$ over a <field> $F$ and returns a <scalar> in $F$. Its defining property is separate <linearity>: holding either input fixed gives a <linear functional> of the other. For <scalars> $a,b$ and <vectors> $u,v,w$,
$$
\begin{aligned}
B(au+bv,w)&=aB(u,w)+bB(v,w),\\
B(u,av+bw)&=aB(u,v)+bB(u,w).
\end{aligned}
$$
This is a scalar-valued <bilinear map>; the chosen <field> is part of the definition.https://ocw.mit.edu/courses/res-18-011-algebra-i-student-notes-fall-2021/mit18_701f21_lect24.pdf[MIT Algebra I, Lecture 24, §24.2: bilinear forms]{ref}
Separate <linearity> does not mean that $B$ is a <linear map> on the <vector space> $V\times V$ with componentwise <vector addition> and <scalar multiplication>. Over the <real numbers>, $B(2v,2w)=4B(v,w)$, whereas a <linear map> on the pair would give $2B(v,w)$. The simplest example is $B(x,y)=xy$ on the one-dimensional <vector space> $\mathbb R$: fixing $y=3$ gives $x\mapsto3x$, but doubling both $x$ and $y$ quadruples their product.
A <bilinear form> measures how two inputs interact. The <dot product> measures alignment; a <symmetric bilinear form> can describe a <quadratic form>; an <alternating bilinear form> can measure signed <area>. A <nondegenerate bilinear form> turns each <vector> into a unique <linear functional> and so connects geometry with the <dual space>. The same short definition therefore supports lengths, <bilinear orthogonal complements>, and <symplectic vector spaces>, depending on the additional properties imposed.
Over the <complex numbers>, a <bilinear form> is <linear> in both inputs without conjugating either <scalar>. A <Hermitian form> instead is a <sesquilinear form> and conjugates when its inputs are exchanged. For complex <matrices>, the adjoint $A^*$ used with <Hermitian forms> and matrix <C-star algebras> is the <conjugate transpose>, whereas a <bilinear form> uses the ordinary <transpose> $A^T$. For example, $B(z,w)=zw$ has $B(i,i)=-1$, while the <Hermitian form> $H(z,w)=z\overline w$ has $H(i,i)=1$.<Bilinear Forms by Keith Conrad>[§1: examples and the Hermitian distinction]{ref}
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