Witt's theorem

Remarks

The usual proof is by induction on $\dim U$, and it builds $\tau$ as a product of reflections in non-isotropic vectors; dividing by $2$ is needed to write these reflections. A consequence is Witt cancellation: if $V_1 \perp V_2$ and $V_1 \perp V_3$ are isometric and $V_1, V_2, V_3$ are nondegenerate, then $V_2$ and $V_3$ are isometric. Another consequence: all maximal totally isotropic subspaces of $V$ have the same dimension, the Witt index. For example, in $\R^n$ with the dot product, any isometric map between two subspaces extends to an orthogonal map of $\R^n$.

The hypothesis on the characteristic is necessary for symmetric forms. Over $\F_2$, take $V = \F_2^3$ with the dot product, $u = (1,1,1)$ and $e_1 = (1,0,0)$; both have $B(x,x) = 1$, so $e_1 \mapsto u$ is an isometry between lines. An extension to $V$ would map the orthogonal complement of $e_1$ onto that of $u$. But the complement of $e_1$ contains $(0,1,0)$ with $B(x,x) = 1$, while every $x$ in the complement of $u$ has $B(x,x) = x_1 + x_2 + x_3 = 0$. The theorem holds in every characteristic for nondegenerate alternating forms and for quadratic forms with nondegenerate polar form.

Going deeper

For a student

Witt's theorem is the foundation of the algebraic theory of quadratic forms. Cancellation reduces the classification of forms to anisotropic forms plus a number of hyperbolic planes, the Witt decomposition. This decomposition leads to the Witt group of a field. For real forms, cancellation gives a short proof of Sylvester's law of inertia.