Page 187 - Matrix Analysis & Applied Linear Algebra

182 Chapter 4 Vector Spaces

It is important to realize that the concepts of linear independence and de-
pendence are defined only for sets—individual vectors are neither linearly inde-
pendent nor dependent. For example consider the following sets:

1 0 1 1 1 0 1
S 1 = , , S 2 = , , S 3 = , , .
0 1 0 1 0 1 1
It should be clear that S 1 and S 2 are linearly independent sets while S 3 is
linearly dependent. This shows that individual vectors can simultaneously belong
to linearly independent sets as well as linearly dependent sets. Consequently, it
makes no sense to speak of “linearly independent vectors” or “linearly dependent
vectors.”
Example 4.3.1

Problem: Determine whether or not the set
     
1 1 5
 
2
,
6
S =      
0
,
1 2 7
 
is linearly independent.
Solution: Simply determine whether or not there exists a nontrivial solution
for the α ’s in the homogeneous equation
       
1 1 5 0
2
0
6
0
α 1   + α 2   + α 3   =  
1 2 7 0
or, equivalently, if there is a nontrivial solution to the homogeneous system
     
115 α 1 0
206 = 0 .
   α 2   
127 α 3 0
1 1 5 1 0 3

If A = 2 0 6 , then E A = 0 1 2 , and therefore there exist nontrivial
1 2 7 0 0 0
solutions. Consequently, S is a linearly dependent set. Notice that one particular
dependence relationship in S is revealed by E A because it guarantees that
A ∗3 =3A ∗1 +2A ∗2 . This example indicates why the question of whether or not
m
a subset of is linearly independent is really a question about whether or not
the nullspace of an associated matrix is trivial. The following is a more formal
statement of this fact.