Page 440 - Advanced Linear Algebra
P. 440
424 Advanced Linear Algebra
2. If ( is an d matrix prove that the set ¸(%% s Á % ¹ is a
convex cone in s .
)
(
3. If and are strictly separated subsets of s and if is finite, prove that
(
( ) and are strongly separated as well.
-
4. Let = be a vector space over a field with char ² - ³ £ . Show that a
=
subset ? of is closed under the taking of convex combinations of any
two of its points if and only if ? is closed under the taking of arbitrary
convex combinations, that is, for all ,
% Á Ã Á % ?Á ~ Á ¬ % ?
~ ~
5. Explain why an ² c ³ -dimensional subspace of s is the solution set of a
linear equation of the form % bÄb % ~ .
6 Show that
À
> > b ²5Á ³ ~ > c ²5Á ³ q ²5Á ³
and that > k , > ²5Á ³ k and ²5Á ³ are pairwise disjoint and
> ²5Á ³
b c
> k b > ²5Á ³ r k c > ²5Á ³ r s ²5Á ³ ~
7. A function ;¢ s ¦ s is affine if it has the form ;²#³ ~ # b for
s , where ² s Á s ³. Prove that if * s is convex, then so is
B
;²*³ s .
8. Find a cone in s that is not convex. Prove that a subset ? of s is a
convex cone if and only if %Á & ? implies that % b & ? for all
Á .
9. Prove that the convex hull of a set ¸% ÁÃÁ% ¹ in s is bounded, without
using the fact that it is compact.
10. Suppose that a vector % s has two distinct representations as convex
. Prove that the vectors
combinations of the vectors #Á Ã Á #
are linearly dependent.
#c # Á Ã Á # c #
11. Suppose that is a nonempty convex subset of s and that ² > 5 Á ³ is a
*
hyperplane disjoint from . Prove that lies in one of the open half-spaces
*
*
determined by >²5Á ³ .
12. Prove that the conclusion of Theorem 15.6 may fail if we assume only that
* is closed and convex.
13. Find two nonempty convex subsets of s that are strictly separated but not
strongly separated.
?
@
14. Prove that and are strongly separated by ² 5 > Á ³ if and only if
Z
Z
Z
º5Á % » for all % ? and º5Á & » for all & @ Z
where ? ~ ? b )² Á ³ and @ ~ @ b )² Á ³ and where )² Á ³ is the
closed unit ball.
