Page 309 - Advanced Linear Algebra

Metric Vector Spaces: The Theory of Bilinear Forms 293

For real inner product spaces, Theorem 10.16 says that if ))#~$£ , then
) )
$
/ # #c$ is the unique reflection sending to , that is, / #c$ ² # ³ ~ $ . In the
present context, we must be careful, since symmetries are defined for
nonisotropic vectors only. Here is what we can say.

Theorem 11.32 Let be a nonsingular orthogonal geometry over a field ,
-
=
(
with char²-³ £ . If "Á #  = are nonisotropic vectors with the same nonzero)
“length,” that is, if
º"Á"» ~ º#Á#» £
then there exists a symmetry for which

or "~# " ~c#

"
#
Proof. Since and are nonisotropic, one of " c # or " b # must also be
nonisotropic, for otherwise, since "c# and " b# are orthogonal, their sum "
would also be isotropic. If "b# is nonisotropic, then
"b# ²" b#³ ~ c²" b#³
and

"b# ²" c #³ ~ " c #
and so "b# "~c# . On the other hand, if " c # is nonisotropic, then
"c# ²" c#³ ~ c²" c#³

and
"c# ²" b #³ ~ " b #

and so "c# "~# .
Recall that an operator on a real inner product space is unitary if and only if it is
a product of reflections. Here is the generalization to nonsingular orthogonal
geometries.

Theorem 11.33 Let be a nonsingular orthogonal geometry over a field -
=
with char²-³ £ . A linear transformation on = is an orthogonal
transformation if and only if is the product of symmetries on .

=
Proof. The proof is by induction on ~ dim ²= ³ . If ~ , then = ~ span ²#³
where º#Á #» £ . Let # ~ # for  - . Since is unitary

º#Á #» ~ º #Á #» ~ º #Á #» ~ º#Á #»

and so . If ~f , then is the identity, which is equal to # . On the
~
other hand, if ~c then ~ # . In either case, is a product of symmetries.