Page 311 - Advanced Linear Algebra
P. 311
Metric Vector Spaces: The Theory of Bilinear Forms 295
Suppose first that dim²:³ ~ and that : ~ span ² ³ . Since
º Á »~º Á »£
Theorem 11.32 implies that there is a symmetry for which ~ where
=
~f . Hence, is an isometry of for which ; ~ : and Theorem 11.9
implies that ; ~ ² : ³ . Thus, O : is the desired isometry.
Now suppose the theorem is true for dim²:³ and let dim²:³ ~ . Let
¢: ¦ ; be an isometry. Since : is nonsingular, we can choose a nonisotropic
<
vector : and write : ~ span ² ³ p < , where is nonsingular. It follows
that
=~ : p : ~ span ² ³ p < p :
and
span
=~ ; p ; ~ ² ² ³³ p < p ;
Now we may apply the one-dimensional case to deduce that
<p : <p ;
If ¢ < p: ¦ < p; is an isometry, then
< p ²: ³~ ²< p : ³~ < p ;
But < < and since dim ² <³ ~ dim ²<³ , the induction hypothesis
implies that : ²: ³ ; .
As we have seen, Witt's extension theorem is a corollary of Witt's cancellation
theorem.
(
Theorem 11.35 Witt's extension theorem) Let = and = Z be isometric
nonsingular orthogonal geometries over a field , with char ² - ³ £ . Suppose
-
that is a subspace of and
=
<
¢< ¦ < = Z
is an isometry. Then can be extended to an isometry from to = Z .
=
Maximal Hyperbolic Subspaces of an Orthogonal Geometry
We have seen that any orthogonal geometry can be written in the form
=
=~ < p rad ²= ³
where is nonsingular. Nonsingular spaces are better behaved than singular
<
ones, but they can still possess isotropic vectors.
