Linear Independence
Motivation:
Let
S  v1 , v2 ,, vk 
and
span ( S )  W
. Is it possible to find a
smaller (or even smallest) set, for example, S  v1 , v2 ,, vk 1 , such

that
span( S )  W  span( S  ) ?
To answer this question, we need to introduce the concept of
linear
independence and linear dependence.
Definition of linear dependence and linear independence:
The vectors
v1 , v2 ,, vk
there exist constants,
in a vector space V are said to linearly dependent if
c1 , c2 ,, ck , not all 0, such that
c1v1  c2v2    ck vk  0 .
v1 , v2 ,, vk
are linearly independent if
c1v1  c2v2    ck vk  0  c1  c2    ck  0 .
The procedure to determine if
v1 , v2 ,, vk
are linearly dependent or linearly
independent:
1. Form
equation
c1v1  c2v2    ck vk  0
,
which
lead
to
a
homogeneous system.
2. If the homogeneous system has only the trivial solution, then the given
vectors are linearly independent; if it has a nontrivial solution, then the
vectors are linearly dependent.
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Example:
1
0
0 
e1  0, e2  1, e3  0, and S  e1 , e2 , e3  . Are e1 , e2 and e3
0
0
1
linearly independent?
[solution:]
1
0
0 1 0 0  c1 
c1e1  c2 e2  c3 e3  c1 0  c2 1  c3 0  0 1 0 c2   0
0
0
1 0 0 1 c3 
 c1
 
c 2

 c3

0 
  0 

 .



0 

Therefore, e1 , e2 and e3 are linearly independent.
Example:
1 
  2
8
v1  2, v2   1 , v3   6 . . Are v1 ,v2 and v3 linearly independent?
3
 1 
10
[solution:]
1
  2
 8  1  2 8   c1 




c1v1  c2 v2  c3v3  c1 2  c2  1   c3  6   2 1 6  c2   0
3
 1 
10 3 1 10 c3 
 c1 
 4 


, t  R
 c2   t 

2
.




 c3 

  1

2
Therefore, v1 ,v2 and v3 are linearly dependent.
Example:
Determine whether the following set of vectors in the vector space consisting of all
2  2 matrices is linearly independent or linearly dependent.
2 1 3 0 1 0 
S  v1 , v2 , v3   
, 2 1, 2 0  .
0
1

 
 


[solution:]
2 1
 3 0
1 0 0 0
c1v1  c2 v2  c3 v3  c1 

c

c
 2 2 1 3 2 0  0 0 .
0 1



 

Thus,
2c1  3c 2  c3  0
0
c1

2c 2  2c3  0
c1  c 2
0
 2
 3
1  0 
1 
0 
 0  0 




c1
 c2
 c3      .
0 
 2
 2  0 
 
 
   
1 
1 
 0  0 
The homogeneous system is
 2 3 1
0 
c


1
1 0 0 
 

 c 2   0 
0 2 2    0  .

 c3   
1
1
0


0 
The associated homogeneous system has only the trivial solution
 c1  0
c    0 
 2   .
c3  0
Therefore, v1 ,v2 and v3 are linearly independent.
Example:
Determine whether the following set of vectors in the vector space consisting of all
3
polynomials of degree  n is linearly independent or linearly dependent.


S  v1 , v2 , v3   x 2  x  2, 2 x 2  x, 3x 2  2 x  2 .
[solution:]

 
 

c1v1  c2 v2  c3 v3  c1 x 2  x  2  c2 2 x 2  x  c3 3x 2  2 x  2  0 .
Thus,
1
 2
3 0
c1 1  c2 1  c3 2  0 .
2
0
2 0
c1  2c 2  3c3  0
c1  c 2  2c3  0
2c1 

 2c3  0
The associated homogeneous system is
1
1

2
2
1
0
3  c1  0
2 c 2   0 .
2 c3  0
The homogeneous system has infinite number of solutions,
 c1 
1
c   t  1 , t  R.
 2
 
c3 
 1
Therefore, v1 ,v2 and v3 are linearly dependent since
tv1  tv2  tv3  0, t  R .
Note:
In the examples with
1
  2
8
v1  2, v2   1 , v3   6 ,
3
 1 
10

or with

S  v1 , v2 , v3   x 2  x  2, 2 x 2  x, 3x 2  2 x  2 , v1 ,v2 and v3 are
linearly dependent. Observe that v3 in both examples are linear
combinations of v1 ,v2 ,
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8
1 
  2




v3   6   42  2 1   4v3  2v 2
10
3
 1 
and

 

v3  3x 2  2 x  2  x 2  x  2  2 x 2  x  v1  v2 .
As a matter of fact, we have the following general result.
Important result:
The nonzero vectors
v1 , v2 ,, vk
in a vector space V are linearly
dependent if and only if one of the vectors
combination of the preceding vectors
v j , j  2 , is a linear
v1 , v2 ,, v j 1 .
Note:
Every set of vectors containing the zero vector is linearly dependent. That
is,
v1 , v2 ,, vk
vector, then
are k vectors in any vector space and v i is the zero
v1 , v2 ,, vk
are linearly dependent.
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