Page 146 - Advanced Linear Algebra

130 Advanced Linear Algebra

and
~ ² b ³² b ³ ~ ² ³

B
which shows that ~ and ~ . Moreover, if  ²= ³ , then we define
and by

² ³ ~ ² ³

² ³ ~ ² b ³
from which it follows easily that
~ b
B
which shows that ¸Á ¹ is a basis for ²= ³ .
More generally, we begin by partitioning 8 into blocks. For each
~ Á Ã Á c , let

~¸ “ – mod ¹
8
Now we define elements ²= ³ by
B

² b ! ³ ! ~ Á
where ! and where !Á is the Kronecker delta function. These functions
9 ~¸ Á Ã Á ¹ is
are surjective and have disjoint support. It follows that 0 c
linearly independent. For if

~ b Ä b c c

where B , then, applying this to ²= ³ b ! gives
~ ² b! ³~ ! ² ³
!!

for all . Hence, ! ~ .
B
B
9
Also, spans ²= , for if  ²= ³ , we define  ²= ³ by
³
B
² ³ ~ ² + ³

to get
² b Ä b c c ³² !+ ³ ~ ! ² !+ ³ ~ ! ² ³ ~ ² !+ ³

!

and so
~ b b Ä c c

9 ~¸ Á Ã Á ¹ is a basis for ²= ³ of size .
B
Thus, 0 c
Recall that if is a basis for a vector space over , then is isomorphic to
=
=
-
)
) of all functions from to that have finite support. A
the vector space ²- ³ ) -