Let \(k\) be a field and \(V\) a \(k\)-vector space. A quadratic form on \(V\) is a function \(q\colon V\to k\) such that \(q(\lambda x) = \lambda^2 q(x)\) for all \(\lambda\in k\) and \(x\in V\), and such that \[ b_q(x, y) \coloneqq q(x+y) - q(x) - q(y) \] is a bilinear form \(V\times V\to k\).
The form \(b_q\) is symmetric and satisfies \(b_q(x,x)=q(2x)-2q(x)=2q(x)\). If \(\characteristic k \neq 2\), then \(q\mapsto \frac12 b_q\) is a bijection from quadratic forms on \(V\) to symmetric bilinear forms on \(V\), with inverse \(b\mapsto \bigl(x\mapsto b(x,x)\bigr)\). If \(V\) has basis \(e_1,\ldots,e_n\), then in the coordinates \(x=\sum_i x_ie_i\) the form \(q\) is a homogeneous polynomial of degree \(2\), \(q(x) = \sum_{i \leq j} a_{ij} x_i x_j\) with \(a_{ii}=q(e_i)\) and \(a_{ij}=b_q(e_i,e_j)\) for \(i<j\).
See [@Art11].
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