Page 115 - Advanced Linear Algebra

The Isomorphism Theorems 99

~ Á 8 ²#³ Á 9
²#³~

and since the inner sums are in and ¸ ¹ is -independent, the inner sums
-
-
must be zero:
²#³ ~
Á

Since this holds for all # 8 , we have
~
Á

~ for all Á . Hence, ¸ ¹ is linearly independent over
which implies that Á
2 ². This proves that dim - - 8 ³  dim 2 2 2 8 . 3
For the center inequality, it is clear that

8
2 8 ²- ³ 3 dim -  ²- ³
dim -
We will show that the inequality must be strict by showing that the cardinality
8 8 . To this end, the
of ²- ³ is ((8 whereas the cardinality of - is greater than ((8
8 can be partitioned into blocks based on the support of the function. In
set ²- ³
particular, for each finite subset of , if we let
8
:
8
(~ ¸  ²- ³ “ supp ² ³ ~ :¹

:
then
8
²- ³ ~ ( :

:‹8
: finite
where the union is disjoint. Moreover, if ((:~ , then
( (( (  : ( -  L
and so

b 8 b²- ³ ~ (( : (  8 (( h L ~ max ² 8 ((Á L ³ ~ 8 ((

:‹8
: finite
But since the reverse inequality is easy to establish, we have
b 8 b²- ³ ~ (( 8

As to the cardinality of - 8 , for each subset of , there is a function ;  - 8
;
8

that sends every element of to and every element of ± ;8 to . Clearly,

;

each distinct subset gives rise to a distinct function ; and so Cantor's
;