Page 284 - Advanced Linear Algebra
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268 Advanced Linear Algebra
\
$
and so ²$ b"³ = , which implies that $ b" is isotropic. Thus, is also
isotropic and so all vectors in are isotropic.
=
Orthogonal and Symplectic Geometries
If a metric vector space is both orthogonal and symplectic, then the form is both
symmetric and skew-symmetric and so
º"Á #» ~ º#Á "» ~ cº"Á #»
Therefore, when char²-³ £ , is orthogonal and symplectic if and only if =
=
is totally degenerate.
However, if char²-³ ~ , then there are orthogonal symplectic geometries that
are not totally degenerate. For example, let =~ span ²"Á #³ be a two-
dimensional vector space and define a form on whose matrix is
=
4~ > ?
Since 4 is both symmetric and alternate, so is the form.
Linear Functionals
The Riesz representation theorem says that every linear functional on a finite-
dimensional real or complex inner product space is represented by a Riesz
=
vector 9 = , in the sense that
²#³ ~ º#Á 9 »
for all #= . A similar result holds for nonsingular metric vector spaces.
Let be a metric vector space over . Let - % = and define the inner product
=
map ºhÁ %»¢ = ¦ - by
ºhÁ %»# ~ º#Á %»
This is easily seen to be a linear functional and so we can define a linear map
¢= ¦ = by
i
%~º h Á %»
The bilinearity of the form ensures that is linear and the kernel of is
ker² ³ ~ ¸% = º= Á %» ~ ¸ ¹¹ ~ = ~ rad ²= ³
Hence, is injective (and therefore an isomorphism) if and only if = is
nonsingular.
(
Theorem 11.5 The Riesz representation theorem) Let = be a finite-
dimensional nonsingular metric vector space. The map ¢= ¦ = i defined by