Page 315 - Advanced Linear Algebra
P. 315
Metric Vector Spaces: The Theory of Bilinear Forms 299
13. If ºÁ » is a symmetric bilinear form on = and char ²-³ £ , show that
8²%³ ~ º%Á %»° is a quadratic form.
=
14. Let be a vector space over a field , with ordered basis ~ - 8 ² # Á Ã Á # . ³
Let ²% Á Ã Á % ³ be a homogeneous polynomial of degree over , that is,
-
a polynomial each of whose terms has degree . The -form defined by
=
-
is the function from to defined as follows. If ~ # #' , then
²#³ ~ ² ÁÃÁ ³
(We use the same notation for the form and the polynomial. ) Prove that -
forms are the same as quadratic forms.
15. Show that is an isometry on if and only if ² 8 # ³ ~ 8 ² # ³ where is
=
8
(
the quadratic form associated with the bilinear form on . Assume that
=
.
char²-³ £ ³
16. Show that a quadratic form on satisfies the parallelogram law:
=
8
8²%b&³ b8²% c&³ ~ ´8²%³ b8²&³µ
17. Show that if is a nonsingular orthogonal geometry over a field , with
-
=
char²-³ £ , then any totally isotropic subspace of = is also a totally
degenerate space.
18. Is it true that = ~ rad ²= ³ p rad ²= ³ ?
=
19. Let be a nonsingular symplectic geometry and let #Á be a symplectic
transvection. Prove that
a ) #Á ~ #Á b
#Á
)
b For any symplectic transformation ,
c ~ #Á
#Á
c ) For - i ,
#Á ~ #Á
d For a fixed #£ , the map ª #Á is an isomorphism from the
)
¸
-
additive group of onto the group #Á - ¹ Sp = ² . ³
%
20. Prove that if is any nonsquare in a finite field - , then all nonsquares
have the form % , for some - . Hence, the product of any two
is a square.
nonsquares in -
21. Formulate Sylvester's law of inertia in terms of quadratic forms on .
=
22. Show that a two-dimensional space is a hyperbolic plane if and only if it is
nonsingular and contains an isotropic vector. Assume that char²-³ £ .
23. Prove directly that a hyperbolic plane in an orthogonal geometry cannot
have an orthogonal basis when char²-³ ~ .
24. a Let < ) be a subspace of = . Show that the inner product
º% b<Á & b<» ~ º%Á &» on the quotient space = °< is well-defined if
and only if < rad ²= . ³
)
b If < rad²= ³ , when is = °< nonsingular?
5
25. Let =~ 5 p : , where is a totally degenerate space.
