We propose to prove Witt's theorem, whose statement is: ``let $F$ and $F'$ be two vector subspaces of $E$ such that there exists an isometry $f$ from $(F, q_{/F})$ to $(F', q_{/F'})$. Then there exists $g \in O(E,q)$ such that $g_{/F} = f$.'' We assume that $F$ and $F'$ are non-singular, with $\operatorname{dim}(F) = \operatorname{dim}(F') = 1$. Let $x \in F$ with $x \neq 0$. Set $y = f(x)$. a) Show that $q(x+y)$ or $q(x-y)$ is non-zero. b) Prove Witt's theorem in this case, using question III.A.2-d).
We propose to prove Witt's theorem, whose statement is: ``let $F$ and $F'$ be two vector subspaces of $E$ such that there exists an isometry $f$ from $(F, q_{/F})$ to $(F', q_{/F'})$. Then there exists $g \in O(E,q)$ such that $g_{/F} = f$.''
We assume that $F$ and $F'$ are non-singular, with $\operatorname{dim}(F) = \operatorname{dim}(F') = 1$. Let $x \in F$ with $x \neq 0$. Set $y = f(x)$.
a) Show that $q(x+y)$ or $q(x-y)$ is non-zero.
b) Prove Witt's theorem in this case, using question III.A.2-d).