Exercise 1, Appendix 0

Let E be a finite-dimensional vector space over a commutative field of characteristic $\neq 2$. Denote by H a nondegenerate quadratic form on E, and by S the symmetric bilinear form on $E \times E$ such that $H(x) = S(x, x)$ for all $x$ in E. Let Q be a quadratic form on E.

a) There exists an endomorphism $u$ of E characterized by the relations $Q(x) = S(u(x), x)$ and $S(u(x), y) = S(x, u(y))$ for all $x$ and $y$ in E. One sets $\operatorname{Tr}(Q/H) = \operatorname{Tr}(u)$.

b) Generalize Remark 3 of No. 1.

c) If $(e_i)_{1 \leq i \leq m}$ is a basis of E orthonormal for H, then $\operatorname{Tr}(Q/H) = \sum_{i=1}^m Q(e_i)$.