Page 130 - Advanced Linear Algebra

114 Advanced Linear Algebra

then let ²?³ ~ % ²?³ b ²?³ where ²?³ does not involve % . This gives

% ~ ´% ²?³ b ²?³µ ²?³

~

²?³ ²?³ b
~ % ²?³ ²?³

~ ~
%

The last sum does not involve and so it must equal . Hence, the first sum

must equal , which is not possible since ² ? ³ has no constant term.
Linear Independence
The concept of linear independence also carries over to modules.
Definition A subset of an -module 4 is linearly independent if for any
9
:
distinct # Á ÃÁ#  : and ÁÃÁ  9 , we have

# bÄb # ~ ¬ ~ for all

A set that is not linearly independent is linearly dependent .
:
It is clear from the definition that any subset of a linearly independent set is
linearly independent.
Recall that in a vector space, a set of vectors is linearly dependent if and only
:
if some vector in is a linear combination of the other vectors in . For
:
:
arbitrary modules, this is not true.
Example 4.3 Consider as a -module. The elements { Á  are linearly
{
{
dependent, since
² ³ c ² ³ ~
(
)
but neither one is a linear combination i.e., integer multiple of the other.
The problem in the previous example as noted earlier is that
(
)
# bÄb # ~

implies that

# ~ c # c Ä c #

, since it may not have a
but in general, we cannot divide both sides by
multiplicative inverse in the ring .
9