Page 99 - Advanced Linear Algebra
P. 99
Linear Transformations 83
=
Theorem 2.29 Let ²= ³ where is a real vector space. The matrix of d
B
8
with respect to the ordered basis cpx²³ is equal to the matrix of with respect
to the ordered basis :
8
´ d µ cpx²³ ~ 8 ´ µ 8
Hence, if a real matrix represents a linear operator on , then also
(
(
=
represents the complexification d of on = d .
Exercises
1. Let ( C have rank . Prove that there are matrices ? C Á Á and
@ C Á , both of rank , for which ( ~ ?@ . Prove that ( has rank if
!
and only if it has the form (~ % & where and are row matrices.
&
%
2. Prove Corollary 2.9 and find an example to show that the corollary does not
hold without the finiteness condition.
3. Let ²= Á > ³ . Prove that is an isomorphism if and only if it carries a
B
basis for to a basis for > .
=
4. If B ²= Á > ³ and B ²= Á > ³ we define the external direct sum
B ^ > ³ by
^ ²= ^ = Á >
² ^ ³²²# Á # ³³ ~ ² # Á # ³
Show that ^ is a linear transformation.
5. Let =~ : l ; . Prove that : l ; : ^ ; . Thus, internal and external
direct sums are equivalent up to isomorphism.
6. Let =~ ( b ) and consider the external direct sum , ~ ( ^ ) . Define a
map ^¢( ) ¦ by ²#Á$³ ~ # b $ . Show that is linear. What is the
=
kernel of ? When is an isomorphism?
7. Let B ² = - ³ where ²dim = ³ ~ B . Let ( C ² - ³ . Suppose that
there is an isomorphism with the property that ² #³ ~ (² #³ .
¢= -
8
Prove that there is an ordered basis for which (~ ´ µ 8 .
J
:
=
:
B
J
8. Let be a subset of ²= ³ . A subspace of is -invariant if is -
J
=
invariant for every J . Also, is -irreducible if the only -invariant
J
=
subspaces of are ¸ ¹ and . Prove the following form of Schur's lemma.
=
Suppose that = J B and > ²= ³ J B ²> ³ and is = J= -irreducible and >
is J -irreducible. Let > B satisfy ²= Á > ³ J = ~ J > , that is, for any
J = J such that ~ > there is a J and for any > there is a
. Prove that
J = such that ~ ~ or is an isomorphism.
9. Let B ²= ³ where dim ²= ³B . If rk ² ³~ rk ² ³ show that
im² ³ q ker ² ³ ~ ¸ ¹.
10. Let ²< ,= ³ and ²= Á > ³ . Show that
B
B
rk ² ³ min rk ³ Á rk ³ ¹
²
²
¸
B
11. Let ²<Á = ³ and ²= Á > ³ . Show that
B
null ² ³ null ³ ² b null ³
²
