A quadratic form over a field $K$ is a map $q:V\to K$ from a finite-dimensional vector space $V$ over $K$ such that
$$ q(av)=a^2 q(v), \,\, a\in K,v\in V $$and the function
$$ b_q(u,v):=\frac{1}{2}(q(u+v)-q(u)-q(v)) $$is bilinear. It takes the form of a degree 2 homogeneous polynomial in $n$ variables with coefficients in $K$.
Fixed a basis of $V$, there exists a $n\times n$ matrix $A$ such that
$$ q(x)=x^T A \,x $$The bilinear map $b_q$ defined above is called the associated bilinear form. It is satisfied that
$$ q(u,v)=v^T A\, u $$and
$$ q(x)=b_q(x,x). $$________________________________________
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Author of the notes: Antonio J. Pan-Collantes
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