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 ~ =  ž  ~ 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
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