Download linearly independent

Linear Equations
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1.1
1.2
1.3
1.4
1.5
1.7
System of linear Equations
Row Reduction and Echelon Forms
Vector Equations
向量方程
The Matrix Equation Ax = b 矩阵方程
Solution Sets of Linear Systems
Linear Independence 线性无关
• Solution
1
 2  7 
x1  2  x2 5   4 
 5
6  3
 1 2 7  1 0 3 
 2 5 4  ~ 0 1 2

 

 5 6 3 0 0 0 
x1  2 x2  7
2 x1  5 x2  4
5 x1  6 x2  3
b = 3a1+ 2a2
x1  2 x2  x3  4
5 x2  3x3  1
1 
2
 1
 4
0 x1   5 x2   3  x3  1 
 
 
 
 
Ax  b
 x1 
1 2 1    4
0 5 3   x2   1 

 x   
 3
a matrix equation
•
a.
b.
c.
d.
THEOREM 4
The following statements are logically
equivalent.
For each b in Rm, the equation Ax = b has
a solution.
Each b in Rm is a linear combination of
the column of A.
The columns of A span Rm.
A has a pivot position in every row.
10 x1  3x2  2 x3  0
 x1 
0.3x2  0.2 x3 
0.3
0.2
  x  1 x  0 
x2
10  3  2  x2   0  x  
 2  3 
 x3 


 0 
 1 
x3
The solution set of Ax=0
can always be expressed explicitly as
Span{v1, v2, … , vp} for suitable vectors v1, v2, … , vp.
1 0 4 / 3 1
[ A, b]  0 1
0
2 
0 0
0
0 
 x1   1  43 x3   1  43 x3   1
 43 
Ax  b  x   x2    2    2    0    2   x3 0
   
1 
 x3   x3   0   x3   0 
• THEOREM 6
Suppose Ax = b is consistent, and let p be a
solution. Then the solution set of Ax = b is the
set of all vectors of the form
w = p + vh ,
where vh is any solution of the homogeneous
equation Ax = 0.
1.7 Linear Independence
线性无关
• 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”
• 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
1 
• Let
 4
 2
v1   2 , v2  5  , v3  1  .
5 
6 
 0 
a. Determine if the set {v1, v2, v3} is linearly
independent.
b. If possible, find a linear dependence relation
among v1, v2, v3.
DOES
x1v1+ x2v2+ x3v3=0
has only the trivial solution?
2 
1 4 2  1 4
[v1 , v2 , v3 ]  2 5 1  ~ 0 3 3 
5 6 0  0 14 10
 1 4 2   1 4 2  1 4 2 
[v1 , v2 , v3 ]  2 5 1  ~ 0 3 3 ~ 0 1 1 
 3 6 0  0 6 6 0 0 0 
1 0 2
2v1-v2+v3=0
~ 0 1 1 
0 0 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 
A1x1+ A2x2+…+ Anxn+=0
• EXAMPLE 2
Determine if the column of the matrix
0 1 4 


A  1 2 -1
5 8 0 
are linearly independent.
1
A ~ 0
5
1
~ 0
0
2 1
1 4 
8 0 
2 1
1 4 
-2 5 
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.
Sets of Two or More Vectors
• 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.
Sets of Two or More Vectors
• 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
Sets of Two or More Vectors
• 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.
x1v1+ x2v2+ … + xpvp=0
[v1 , v2 ,

, v p ]  





• THEOREM 9
• If a set S={v1, v2,…, vp} in Rn contains the zero
vector, then the set is linearly dependent.
50+ x1v1+ x2v2+ … + xpvp=0
• Homework: Exercise1.7 2, 6, 12, 32