Download linearly independent

1.7 Linear Independence
线性无关
THEOREM 4
The following statements are logically equivalent.
a. For each b in Rm, the equation Ax = b has a
solution.
b. Each b in Rm is a linear combination of the
column of A.
c. The columns of A span Rm.
d. A has a pivot position in every row.
1
 2  7 
x1  2  x2 5   4 
 5
6  3
 x1  2 x2  7
1
 2
7

 2 x1  5 x2  4 1   2 ,  2  5 ,  3   4 
 
 
 
 5 x  6 x  3
1
2

 5
6
 3
1 0 3 
1 2 7
 2 5 4   0 1 2 




0 0 0 
 5 6 3
3  31  2 2  31  2 2  3  0
 x1   3 
   

x11  x2 2  x33  0
 x2    2 
 x   1
 3  
1
 2
1 
1
 2
1 
 
 
 
x1  2   x2 5   x3  4  0 1   2  ,  2  5  ,  3   4
 1
1 
 3
 1
1 
 3
1 2 1 
 1 2 1
 2 5 4   0 1 2 




0 0 2
 1 1 3
 x1   0 
   
  x2    0 
 x  0
 3  
x11  x2 2  x33  0  x1  x2  x3  0
DEFINITION
• A set {v1, v2, … , vp} in Rn is said to be linearly
independent
if the vector equation
x1v1+ x2v2+ … + xpvp=0
has only the trivial solution.
• The set {v1, v2, … , vp} is said to be linearly
dependent
if there exist weights c1, c2, …, cp, not all zero,
such that
c1v1+ c2v2+ … + cpvp=0.
• A set is linearly dependent if and only if it is not
linearly independent.
• 不是“线性独立”
• 没有“线性有关”这个词，应该是“线性相关”
1
 2
7
1   2 ,  2  5 , 3   4 
 5
6
 3
1
 2
1 
1   2  ,  2  5  ,  3   4
 1
1 
 3
linearly dependent
linearly independent
• Linear dependence relation, 线性相关关系
c1v1+ c2v2+ … + cpvp=0
• linearly dependent set, 线性相关组
{v1, v2, … , vp} is linearly dependent
• linearly independent set 线性无关组
{v1, v2, … , vp} is linearly independent
EXAMPLE 1
Let
1 
 4
 2
v1   2 , v2  5  , v3  1  .
5 
6 
 0 
1.Determine if the set {v1, v2, v3} is linearly independent.
2.If possible, find a linear dependence relation among v1, v2, v3.
DOES x1v1+ x2v2+ x3v3=0 has only the trivial solution?
 1 4 2  1 4
2  1 4 2  1 4 2 
0 1 1 







[v1 , v2 , v3 ]  2 5 1   0 3 3  0 1 1

 

 
5 6 0  0 14 10 0 7 5  0 0 2
linearly independent
1 
 4
 2
v1   2 , v2  5  , v3  1  .
 3
 6 
 0 
1 4 2 1 4 2  1 4 2 1 0 2 
0 3 3  0 1 1   0 1 1 



[v1 , v2 , v3 ]  2 5 1  
 

 
 3 6 0  0 6 6 0 0 0  0 0 0 
 x1  2 x3

 x2   x3
 x1   2 
   
  x2    1 x3 .
x   1 
 3  
linearly dependent
2v1-v2+v3=0
Linear Independence of Matrix Columns
The columns of a matrix A are linearly independent
if and only if the equation
Ax=0
has only the trivial solution.
Ax=0  x11  x2 2 
 xn n  0
EXAMPLE 2
Determine if the column of the matrix
0 1 4 
A  1 2 -1
5 8 0 
are linearly independent.
1 2 1
1 2 1
1 2 1
A  0 1 4   0 1 4   0 1 4 
5 8 0 
0  5 
0 0 
Sets of One or Two Vectors
• A set containing only one vector – say, v – is
linearly independent if and only if v is not the
zero vector.
• A set of two vectors {v1, v2} is linearly
dependent if at least one of the vectors is a
multiple of the other. The set is linearly
independent if and only if neither of the
vectors is a multiple of the other.
THEOREM 7
Characterization of Linearly Dependent Sets
A set S={v1, v2,…, vp} of two or more vectors is
linearly dependent if and only if at least one of
the vectors in S is a linear combination of the
others.
• A set S={v1, v2,…, vp} of two or more vectors
is linearly dependent if and only if at least one
of the vectors in S is a linear combination of
the others.
：The set {v1, v2, … , vp} is linearly dependent ,
so there exist weights c1, c2, …, cp, not all zero,
such that c1v1+ c2v2+ … + cpvp=0.
c j  0, c j v j  c1v1  c2 v2 
vj  
c1
c
v1  2 v2 
cj
cj

c j 1
cj
 c j 1v j 1  c j 1v j 1 
v j 1 
c j 1
cj
v j 1 

 c pv p
cp
cj
vp
THEOREM 7
Characterization of Linearly Dependent Sets
A set S={v1, v2,…, vp} of two or more vectors is
linearly dependent if and only if at least one of
the vectors in S is a linear combination of the
others. If S is linearly dependent and v10,then
some vj is a linear combination of the preceding
vectors, v1, v2,…, vj-1.
THEOREM 8
• If a set contains more vectors than there are
entries in each vector, then the set is linearly
dependent.
• That is, any set {v1, v2,…, vp} in Rn is linearly
dependent if p>n.
THEOREM 9
• If a set S={v1, v2,…, vp} in Rn contains the zero
vector, then the set is linearly dependent.