Page 56 - Advanced Linear Algebra
P. 56
40 Advanced Linear Algebra
is called a complete lattice . The least upper bound of a collection is also called
the join of the collection and the greatest lower bound is called the meet .
=
Theorem 1.3 The set I²= ³ of all subspaces of a vector space is a complete
=
lattice under set inclusion, with smallest element ¸ ¹ , largest element , meet
glb¸: 2¹ ~ :
2
and join
lub¸: 2¹ ~ :
2
Direct Sums
As we will see, there are many ways to construct new vector spaces from old
ones.
External Direct Sums
be vector spaces over a field . The external direct
Definition Let =Á Ã Á = -
, denoted by
sum of =Á Ã Á =
Ä ^ = ^
=~ =
is the vector space whose elements are ordered -tuples:
=
= ~ ¸²# Áà Á# ³ # = Á ~ Áà Á ¹
with componentwise operations
²" ÁÃ Á" ³ b ²# ÁÃ Á# ³ ~ ²" b # ÁÃ Á" b # ³
and
²# Á Ã Á # ³ ~ ² # Á Ã Á # ³
for all - .
Example 1.4 The vector space - is the external direct sum of copies of ,
-
that is,
-~ - ^ Ä ^ -
where there are summands on the right-hand side.
This construction can be generalized to any collection of vector spaces by
generalizing the idea that an ordered -tuple ² # Á Ã Á # ³ is just a function
¢ ¸ Á Ã Á ¹ ¦ from the index set ¸ Á Ã Á ¹ to the union of the spaces
=
.
with the property that ² ³ =
