Page 312 - Advanced Linear Algebra
P. 312
296 Advanced Linear Algebra
We can improve upon the preceding decomposition by noticing that if "< is
isotropic, then Theorem 11.10 implies that span²"³ can be “captured” in a
hyperbolic plane /~ span ²"Á %³ . Then we can write
=~ / p / < p rad ²= ³
where / < is the orthogonal complement of / in < and has “one fewer”
isotropic vector. In order to generalize this process, we first discuss maximal
totally degenerate subspaces.
Maximal Totally Degenerate Subspaces
Let be a nonsingular orthogonal geometry over a field , with char ² - ³ £ .
=
-
Suppose that < and < Z are maximal totally degenerate subspaces of = . We
Z
Z
claim that dim²<³ ~ dim²< ³ . For if dim²<³ dim²< ³ , then there is a vector
space isomorphism ¢< ¦ < < Z , which is also an isometry, since and
<
Z
< are totally degenerate. Thus, Witt's extension theorem implies the existence
Z
of an isometry that extends . In particular, ¢= ¦ = c ²< ³ is a totally
degenerate space that contains < and so c ² < Z ³ ~ < , which shows that
Z
dim²<³ ~ dim²< ³.
Theorem 11.36 Let be a nonsingular orthogonal geometry over a field ,
-
=
with char²-³ £ .
)
1 All maximal totally degenerate subspaces of have the same dimension,
=
which is called the Witt index of and is denoted by ² $ = . ³
=
)
=
2 Any totally degenerate subspace of of dimension ² $ = ³ is maximal.
Maximal Hyperbolic Subspaces
We can prove by a similar argument that all maximal hyperbolic subspaces of =
have the same dimension. Let
> ~ / pÄp/
and
A ~ 2 pÄp2
be maximal hyperbolic subspaces of and suppose that / ~ ² " Á # span ³ and
=
2~ span ²% Á & ³ . We may assume that dim ² ³ dim> ² ³.A
The linear map > ¦ ¢ A defined by
"~ % Á #~ &
is clearly an isometry from to > . Thus, Witt's extension theorem implies the
>
existence of an isometry that extends . In particular, ¢= ¦ = c ² ³ is a
A
A
hyperbolic space that contains and so c A > ²³ ~ . It follows that dim ²³
>
~ ² dim > .
³
