 3
1
0






 2   3 0   2 1 
 3
1
0






 3
1

is a linear combination of

 2
 0
 3
1




 3


 2  depends upon
 3


1


 0
1


and
and
 0


1
 0


 0


1
 0


 3 
1
 0 




 is called a DEPENDENT set.

 2 
0
 1 
 3  ,  1  ,  0  






When one vector in a set is a linear combination of other vectors
in the set, then the set is said to be DEPENDENT.
Suppose
w, x, y, z 
is dependent. One of the vectors,
x , is a linear combination of the others:
x  aw  by  cz
Let’s say
x  aw  by  cz  0
A nontrivial linear combination of the vectors produces the zero vector.
At least one coefficient is not 0.
When one vector in a set is a linear combination of other vectors
in the set, then the set is said to be DEPENDENT.
Suppose
w, x, y, z 
is dependent. One of the vectors,
x , is a linear combination of the others:
x  aw  by  cz
Let’s say
x  aw  by  cz  0
A nontrivial linear combination of the vectors produces the zero vector.
At least one coefficient is not 0.
Is the set
 1   0   0  
      
 0 ,  1 ,  0  
 0   0   1  
      
dependent?
1
 0
 0
 0
 
 
 
 
c1  0   c2  1   c3  0    0 
 0
 0
1
 0
 
 
 
 
Is the set
 1   0   0  
      
 0 ,  1 ,  0  
 0   0   1  
      
1
 
0  0  0
 0
 
 0
 
1  0
 0
 
dependent?
 0
 0
 
 
 0   0
1
 0
 
 
This is TRIVIAL. All coefficients are 0’s.
Is the set
 1   0   0  
      
 0 ,  1 ,  0  
 0   0   1  
      
dependent?
NO
1
 0
 0
 0
 
 
 
 
c1  0   c2  1   c3  0    0 
 0
 0
1
 0
 
 
 
 
Is there any way to do this without using ALL ZEROS?
NO
 1   0   0  
      
 0 ,  1 ,  0   is an INDEPENDENT SET
 0   0   1  
      
definition:
v1 , v2 , v3 ,....vn 
is an INDEPENDENT set iff
c1v1  c2 v2  c3v3  ....  cn vn  0
ONLY IF
c1  c2  c3  .......  cn  0
The ONLY linear combination of the vectors to produce the zero
vector is the TRIVIAL one.
Suppose one of these is not 0.
Let’s say c2  0
c1v1  c2 v2  c3v3  ....  cn vn  0
c2 v2  c1v1  c3v3  ....  cn vn
c3
cn
c1
v2   v1  v3  ....  vn
c2
c2
c2
v2
DEPENDS on the other vectors in the set!
 1   1   2  
 
 


Is the set 1 ,  2 ,  5   independent?
  1   1   2  
 
 


 1 
 1 
 2 
 0






 
c1  1   c2  2   c3  5    0 
  1
  1
  2
 0






 
1c1
 1c 2
 2c3
0
1c1
 2c 2
 5c3
0
 1c1
 1c 2
 2c3
0
 1   1   2  
 
 


 1 ,  2 ,  5  
  1   1   2  
 
 


Is the set
independent?
 1 
 1 
 2 
 0








c1  1   c2  2   c3  5    0 
  1
  1
  2
 0








1c1
 1c 2
 2c3
 0
1c1
 2c 2
 5c 3
 0
 1c1
 1c 2
 2c3
 0
2 0
1 1


 1 2 5 0
 1 1  2 0


Reduces
to
1 0 1 0


0 1 3 0
0 0 0 0


 1   1   2  
 
 


 1 ,  2 ,  5  
  1   1   2  
 
 


Is the set
independent?
 1 
 1 
 2 
 0








c1  1   c2  2   c3  5    0 
  1
  1
  2
 0








1c1
 1c 2
 2c3
 0
1c1
 2c 2
 5c 3
 0
 1c1
 1c 2
 2c3
 0
c1 = 1
c1 = c 3
c2 = -3
c2 = -3c3
Let c3 = 1
1 0 1 0


0 1 3 0
0 0 0 0


 1   1   2  
 
 


 1 ,  2 ,  5  
  1   1   2  
 
 


NO
independent?
Is the set
 1 
 1 
 2 
 0






 
c
11  1   c
-3 2  2   c
1 3 5    0
  1
  1
  2
 0






 
1c1
 1c 2
 2c3
 0
1c1
 2c 2
 5c 3
 0
 1c1
 1c 2
 2c3
 0
c1 = 1
c1 = c 3
c2 = -3
c2 = -3c3
Let c3 = 1
1 0 1 0


0 1 3 0
0 0 0 0


 1   1   2  
 
 


 1 ,  2 ,  5  
  1   1   2  
 
 


NO
independent?
Is the set
 1 
 1 
 2 
 0






 
c
11  1   c
-3 2  2   c
1 3 5    0
  1
  1
  2
 0






 
 1 
 1 
 2 






c11  1   c
32  2   c
-13  5 
  1
  1
  2






 1   1   2  
 
 


Is the set 1 ,  2 ,  5   independent?
  1   1   3  
 
 


 1 
 1 
 2 
 0






 
c1  1   c 2  2   c3  5    0 
  1
  1
  3
 0






 
1c1
 1c 2
 2c3
0
1c1
 2c 2
 5c3
0
 1c1
 1c 2
 3c3
0
 1 


 1 ,
  1


Is the set
 1 


c1  1   c 2
  1


 1 


 2 ,
  1


2 



5


  3 


 1 


 2   c3
  1


independent?
2 

 0




5    0

  3
 0




1c1
 1c 2
 2c 3
 0
1c1
 2c 2
 5c 3
 0
 1c1
 1c 2
 3c 3
 0
 1 1 2 0


 1 2 5 0
 1 1  3 0


Reduces
to
1 0 0 0


 0 1 0 0
 0 0 1 0


 1 


 1 ,
  1


Is the set
 1 


 2 ,
  1


2 



5


  3 


independent?
YES
 1 
 1 
 2 
 0






 
c1  1   c 2  2   c3  5    0 
  1
  1
  3
 0






 
1c1
 1c 2
 2c 3
 0
1c1
 2c 2
 5c 3
 0
 1c1
 1c 2
 3c 3
 0
ONLY IF
c1 = 0
c2 = 0
c3 = 0
1 0 0 0


 0 1 0 0
 0 0 1 0

