Page 287 - Advanced Linear Algebra
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Metric Vector Spaces: The Theory of Bilinear Forms 271
i
dim²: ³ b dim²: ³ ~ dim²= ³
i
However, dim²: ³ ~ dim²:³ and so part 1) follows. For part 2), since
rad²:³ ~ : q : : q :~ rad²: ³
the nonsingularity of : implies the nonsingularity of . Then part 1) implies
:
that
dim²:³ b dim²: ³ ~ dim²= ³
and
dim²: ³ b dim²: ³ ~ dim²= ³
Hence, : ~ : and rad ² : ³ ~ ² rad : . ³
The previous theorem cannot in general be strengthened. Consider the two-
dimensional metric vector space =~ span²"Á #³ where
º"Á "» ~ Á º"Á #» ~ Á º#Á #» ~
If :~ span ²"³ , then : ~ span ²#³ . Now, is nonsingular but : is singular
:
and so 2c) does not hold. Also, rad²:³ ~ ¸ ¹ and rad²: ³ ~ : and so 2b)
fails. Finally, : ~ = £ : and so 2a) fails.
Isometries
We now turn to a discussion of structure-preserving maps on metric vector
spaces.
Definition Let and > be metric vector spaces. We use the same notation Á º »
=
for the bilinear form on each space. A bijective linear map ¢= ¦ > is called
an isometry if
º"Á #» ~ º"Á #»
for all vectors and in . If an isometry exists from to > , we say that =
=
"
=
#
and > are isometric and write = > . It is evident that the set of all
isometries from to forms a group under composition.
=
=
If = is a nonsingular orthogonal geometry, an isometry of = is called an
orthogonal transformation. The set E²= ³ of all orthogonal transformations
on is a group under composition, known as the orthogonal group of .
=
=
If = is a nonsingular symplectic geometry, an isometry of = is called a
symplectic transformation. The set Sp²= ³ of all symplectic transformations on
= is a group under composition, known as the symplectic group
= of .
