Page 74 - Advanced Linear Algebra

58 Advanced Linear Algebra

of -tuples of independent vectors in this subspace is

² c ³² c ³Ä² c c ³
c) Show that

5² Á ³ ~ :² Á ³² c ³² c ³Ä² c c ³
How does this complete the proof?
22. Prove that any subspace of s is a closed set or, equivalently, that its set
:

complement :~ s ± : is open, that is, for any %  : there is an open

%

ball )²%Á ³ centered at with radius € for which )²%Á ³ ‹ : .
23. Let 8 and 9 ~¸ Á Ã Á ¹ ~¸ Á Ã Á ¹ be bases for a vector space .
=

Let   c . Show that there is a permutation of ¸ Á à Á ¹ such

that
Á ÃÁ Á ² b ³ ÁÃÁ ² ³
and
Á ÃÁ ² ³ ² ³ Á b ÁÃÁ
are both bases for . Hint : You may use the fact that if 4 is an invertible
=
d matrix and if   , then it is possible to reorder the rows so
that the upper left d submatrix and the lower right ² c ³ d² c ³
submatrix are both invertible. This follows, for example, from the general
(
Laplace expansion theorem for determinants.)
24. Let = be an -dimensional vector space over an infinite field - and

are subspaces of with dim ²: ³   . Prove
=
suppose that :Á Ã Á :
that there is a subspace ; of = of dimension c for which
; q : ~ ¸ ¹ for all .

25. What is the dimension of the complexification = d thought of as a real
vector space?
(
)
26. When is a subspace of a complex vector space a complexification? Let =
be a real vector space with complexification = d and let be a subspace of
<
= d : . Prove that there is a subspace of for which
=
d
<~ : ~ ¸ b ! “ Á !  :¹
<
if and only if is closed under complex conjugation ¢ = d ¦ = d defined
by ²" b #³ ~ " c # .