Download Linear Independence

1.7: Linear Independence
We will study homogeneous equations from a
different perspective by writing them as vector
equations.
 
In section 1.5 we focused on the unknown solutions to Ax  0
In section 1.7 will focus on the vectors
  that appear in the
vector equation corresponding to Ax  0 .
 
Consider the matrix equation, Ax  0 , where
 3 5  4
A   3  2 4 
 6 1  8
 3 5  4  x1  0
So  3  2 4    x2   0 has the equivalent vector

    
 6 1  8  x3  0
equation form:
3
5
 4 0
x1  3  x2  2  x3  4   0
 6 
 1 
  8 0
Note: When x1  x2  x3  0 we have the trivial
solution. Are there other solutions?
Definitions
v , v ,, v  in
1
2
p
 n is said to be linearly independent if
x1 v1  x2 v2    x p v p  0 has only the trivial solution.
v , v ,, v 
1
2
p
in  is said to be linearly dependent if
there exist weights
n
c1 , c2 ,  , c p , not all zero, such that
c1 v1  c2 v2    c p v p  0
Linear Independence

v , v ,, v 
1
2
p
n

in
are linearly independent
x1 v1  x2 v2    x p v p  0 has only the trivial solution.

x1 v1  x2 v2    x p v p  0  x1  x2    x p  0
Examples: Determine each of the following sets of vectors
are linearly independent.
 1
v2   
 0
1.
1 
v1   
 2
2.
1
v1   
0
  2
v2   
 0
3.
1
v1   
 2
 1
v2   
 0
0
v3   
1 
Find a linear dependence relationship among the
vectors:
1
 1
v1    v2   
 2
 0
0
v3   
1 
The columns of a matrix A are linearly independent
 
Ax  0 has only the trivial solution.
Examples: Determine if the columns of the following matrices
are linearly independent.
1 2
1. 

3
4


 1  2
2. 


2
4


Tips to determine the linear dependence
A set of vectors are linearly dependent if any of the following
are true:
1. The set has two vectors and one is a multiple of the other.
2. The set has two or more vectors and one of the vectors is a
linear combination of the others.
3. The set contains more vectors than the number of entries in
each vector.
4. The set contains the zero vector.