Page 316 - Advanced Linear Algebra
P. 316
300 Advanced Linear Algebra
)
a Prove that 5~ rad²= ³ if and only if is nonsingular.
:
)
b If is nonsingular, prove that : = ° rad = ² . ³
:
26. Let dim²= ³ ~ dim²> ³ . Prove that = °rad ²= ³ > °rad ²> ³ implies
= > .
27. Let =~ : p ; . Prove that
a ) rad²= ³ ~ rad²:³ p rad²;³
)
b = ° rad²= ³ :° rad²:³ p ;° rad²;³
c ) dim ² rad = ² ³ ³ ~ dim ² rad : ² ³ ³ b dim ² rad ; ² ³ ³
d = ) is nonsingular if and only if and are both nonsingular.
:
;
28. Let = be a nonsingular metric vector space. Because the Riesz
representation theorem is valid in , we can define the adjoint of a linear
i
=
map B ²= ³ exactly as in the case of real inner product spaces. Prove
(
that is an isometry if and only if it is bijective and unitary that is,
i
~ ).
B
29. If char²-³ £ , prove that ²= Á > ³ is an isometry if and only if it is
bijective and º Á » ## ~ º # Á # » for all # = .
30. Let 8 be a basis for = . Prove that ~¸# Á Ã Á # ¹ ²= Á > ³ is an
B
isometry if and only if it is bijective and º # Á # » ~ º# Á# » for all Á .
31. Let be a linear operator on a metric vector space . Let 8 ~ ²# ÁÃÁ# ³
=
be an ordered basis for and let 4 8 be the matrix of the form relative to
=
8 . Prove that is an isometry if and only if
!
8
8
´µ 4´µ ~ 4 8
8
32. Let = be a nonsingular orthogonal geometry and let B ² = ³ be an
isometry.
a) Show that dim ker² c³³ ~ dim² im² c³ . ³
²
)
b Show that ker²c ³ ~ im²c ³ . How would you describe
ker²c ³ in words?
c If is a symmetry, what is dim ker c ² ³ ? ³
)
²
)
d Can you characterize symmetries by means of dim ker c ² ³ ? ³
²
B
33. A linear transformation ²= ³ is called unipotent if c is nilpotent.
Suppose that is a nonisotropic metric vector space and that is unipotent
=
and isometric. Show that ~ .
<
34. Let be a hyperbolic space of dimension and let be a hyperbolic
=
=
subspace of of dimension . Show that for each , there is a
hyperbolic subspace > of for which < > = .
=
35. Let char²-³ £ . Prove that if ? is a totally degenerate subspace of an
orthogonal geometry , then dim ² ? ³ ² = ³ ° . dim
=
36. Prove that an orthogonal geometry of dimension is a hyperbolic space
=
if and only if = is nonsingular, is even and = contains a totally
degenerate subspace of dimension ° .
37. Prove that a symplectic transformation has determinant equal to .
