Page 291 - Advanced Linear Algebra

Metric Vector Spaces: The Theory of Bilinear Forms 275

<
3 ; ) is a nonsingular extension of and
dim²;³ ~ dim²<³ b dim²rad ²<³³

Thus, any two nonsingular completions of are isometric.
<
Proof. If < ?  = where ? is nonsingular, then we may apply Theorem
11.10 to as a subspace of , to obtain a hyperbolic extension A p > of <
<
?
for which
<‹ A p > ‹ ?
Thus, every nonsingular extension of contains a hyperbolic extension of .
<
<
Moreover, all hyperbolic extensions of have the same dimension:
<
dim ² p> > ³ ~ dim < ² ³ b dim rad ² < ³ ³
²
and so no hyperbolic extension of is properly contained in another hyperbolic
<
extension of < . This proves that 1)–3) are equivalent. The final statement
follows from the fact that hyperbolic spaces of the same dimension are
isometric.
Extending Isometries to Nonsingular Completions

Let = and = Z be isometric nonsingular metric vector spaces and let
<~ rad ²<³ p > be a subspace of = , with nonsingular completion
<~ > p > .

If ¢< ¦ < is an isometry, then it is a simple matter to extend to an

isometry from < onto a nonsingular completion of < . To see this, let
²" Á' ÁÃÁ" Á' ³ be a hyperbolic basis for . Since > ²" Á ÃÁ" ³ is a basis for

rad²<³, it follows that ² " Á Ã Á " ³ is a basis for rad² <³.

Hence, we can hyperbolically extend <~ rad ² > ³ p > to get

><~ Z p >
ÁÃÁ " Á% ³
where > Z has hyperbolic basis ² " Á% . To extend , simply set
for all ~ ÁÃÁ .
' ~ %
Theorem 11.12 Let and = Z be isometric nonsingular metric vector spaces
=
and let be a subspace of , with nonsingular completion . Any isometry
<
<
=
¢< ¦ < can be extended to an isometry from < onto a nonsingular
completion of < .
The Witt Theorems: A Preview
There are two important theorems that are quite easy to prove in the case of real
inner product spaces, but require more work in the case of metric vector spaces
in general. Let and = Z be isometric nonsingular metric vector spaces over a
=
field . We assume that char ² - ³ £ if is orthogonal.
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=