Page 314 - Advanced Linear Algebra
P. 314
298 Advanced Linear Algebra
4. Show that a metric vector space is nonsingular if and only if the matrix
=
4 8 of the form is nonsingular, for every ordered basis .8
5. Let be a finite-dimensional vector space with a bilinear form Á º » . We do
=
not assume that the form is symmetric or alternate. Show that the following
are equivalent:
)
a ¸# = º#Á $» ~ for all $= ¹~
)
b ¸# = º$Á #» ~ for all $= ¹~
Hint : Consider the singularity of the matrix of the form.
6. Find a diagonal matrix congruent to
v y
w c z
7. Prove that the matrices
0~ > ? and 4 ~ > ?
are congruent over the base field -~ r of rational numbers. Find an
invertible matrix such that 7 ! 0 7 ~ 4 .
7
8. Let be an orthogonal geometry over a field with char ² - ³ £ . We
=
-
=
wish to construct an orthogonal basis E ~²" Á Ã Á " ³ for , starting with
any generating set = ~²# Á Ã Á # ³ . Justify the following steps, essentially
due to Lagrange. We may assume that is not totally degenerate.
=
)
a If º# Á # » £ for some , then let " ~ # . Otherwise, there are indices
£ for which º# Á # » £ . Let " ~ # b # .
)
b Assume we have found an ordered set of vectors E ~²" Á Ã Á " ³
=
=
that form an orthogonal basis for a subspace of and that none of
the 's are isotropic. Then = ~ = p = .
"
)
c For each # = , let
º# Á " »
$~ # c "
~ º" Á " »
Then the vectors $ span = . If = is totally degenerate, take any
)
basis for = and append it to E . Otherwise, repeat step a on = to
get another vector " b and let E b ~ ²" ÁÃÁ" b ³ . Eventually, we
for .
=
arrive at an orthogonal basis E
9. Prove that orthogonal hyperbolic planes may be characterized as two-
dimensional nonsingular orthogonal geometries that have exactly two one-
)
(
dimensional totally isotropic equivalently: totally degenerate subspaces.
10. Prove that a two-dimensional nonsingular orthogonal geometry is a
hyperbolic plane if and only if its discriminant is -² c ³ .
11. Does Minkowski space contain any isotropic vectors? If so, find them.
12. Is Minkowski space isometric to Euclidean space s ?
