Page 119 - Advanced Linear Algebra

The Isomorphism Theorems 103

:b ; ‹ ²: q ;³
For the reverse inclusion, suppose that  ²: q ;³ . Write
Z
Z
=~ : l ²: q ;³ l ; l <
Z
where :~ : l ²: q ;³ and ; ~ ²: q ;³ l ; Z . Define  = i by
O ~ Á O : q ; ~ O : q ; ~ Á O ; Z ~ Á O ~
:
<
Z
and define  = i by
O ~ Á O : q ; ~ O : q ; ~ Á O ; Z ~ Á O ~
Z
:
<
It follows that  ; ,  : and b ~ .
Annihilators and Direct Sums
Consider a direct sum decomposition

=~ : l ;

Then any linear functional  ; i can be extended to a linear functional on =

by setting ²:³ ~ . Let us call this extension by . Clearly,  : and it is

easy to see that the extension by map ¦ is an isomorphism from ; i to
: ; , whose inverse is the restriction to .
Theorem 3.15 Let =~ : l ; .
a The extension by map is an isomorphism from ; i to : and so
)

i
;š :
)
b If is finite-dimensional, then
=

dim²: ³ ~ codim = ²:³ ~ dim²= ³ c dim²:³
Example 3.5 Part b of Theorem 3.15 may fail in the infinite-dimensional case,
)

since it may easily happen that :š = i . As an example, let be the vector
=
space over { with a countably infinite ordered basis 8 ~ ² Á Á Ã ³ . Let

i
i
: ~ º » and ; ~ º Á Á à ». It is easy to see that : š ; š = and that

i
dim²= ³ € dim²= ³.
The annihilator provides a way to describe the dual space of a direct sum.

Theorem 3.16 A linear functional on the direct sum =~ : l ; can be written
as a sum of a linear functional that annihilates and a linear functional that
:
annihilates , that is,
;

i
²: l ;³ ~ : l ;