= Quadratic form
{title2=$Q(av)=a^2Q(v)$}
{wiki}
A <quadratic form> on a <vector space> over a <field> $F$ is a function $Q:V\to F$ satisfying $Q(av)=a^2Q(v)$ whose <polar bilinear form of a quadratic form> $B_Q(v,w)=Q(v+w)-Q(v)-Q(w)$ is a <bilinear form>. In finite <dimension>, it is represented in <coordinates> by a homogeneous degree-two <polynomial>, or by $Q(v)=v^TAv$ with a suitable <matrix> $A$.
When the <characteristic> is not two, one may choose $A$ to be a <symmetric matrix> and recover it from $B_Q/2$. In <characteristic> two, the symmetric-matrix restriction would lose examples: $Q(x,y)=xy$ has an upper-triangular representing <matrix> $\begin{pmatrix}0&1\\0&0\end{pmatrix}$, while $v^TAv$ for a <symmetric matrix> has no $xy$ cross term. Its polar form is $B_Q((x,y),(u,v))=xv+yu$, an <alternating bilinear form>. The distinction matters when studying <quadratic forms> over <finite fields>.<Bilinear Forms by Keith Conrad>[§§6–7: quadratic forms and characteristic two]{ref}
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