Page 288 - Advanced Linear Algebra
P. 288
272 Advanced Linear Algebra
Note that, in contrast to the case of real inner product spaces, we must include
the requirement that be bijective since this does not follow automatically if =
is singular. Here are a few of the basic properties of isometries.
Theorem 11.9 Let B ²= Á > ³ be a linear transformation between finite-
dimensional metric vector spaces and > .
=
)
8
=
1 Let ~¸# Á Ã Á # ¹ be a basis for . Then is an isometry if and only if
is bijective and
º# Á # » ~ º# Á # »
for all Á .
)
=
2 If is orthogonal and char - ² ³ £ , then is an isometry if and only if it is
bijective and
º#Á #» ~ º#Á #»
for all #= .
)
3 Suppose that ¢= > is an isometry and
=~ : p : and > ~ ; p ;
If , then :~ ; ²: ³ ~ ; .
)
Proof. We prove part 3 only. To see that ²: ³ ~ ; , if ' : and ! ; ,
then since ;~ : , we can write ! ~ for some : and so
º 'Á !»~º 'Á »~º'Á »~
whence ²: ³ ; . But since the dimensions are equal, it follows that
²: ³ ~ ; .
Hyperbolic Spaces
A special type of two-dimensional metric vector space plays an important role in
the structure theory of metric vector spaces.
Definition Let = be a metric vector space. A hyperbolic pair is a pair of
vectors "Á # = for which
º"Á"» ~ º#Á#» ~ Á º"Á#» ~
Note that º#Á"» ~ if is orthogonal and º#Á"» ~ c if is symplectic. In
=
=
either case, the subspace /~ span ²"Á #³ is called a hyperbolic plane and any
space of the form
> ~ / pÄp/
where each / is a hyperbolic plane, is called a hyperbolic space . If " ² Á # ³ is
, then we refer to the basis
a hyperbolic pair for /
²" Á# ÁÃÁ" Á# ³
