Chapter 3---Section 3 Linear Independence
?1???2???1???????x1??1 x2?3 x3?3???????8??2??1???????
These three vectors satisfy
(1)
x3=3x1+2x2
Any linear combination of combination of (2)
x1,x2x1,x2,x3 can be reduced to a linear
. Thus S= Span(x1,x2,x3)=Span(x1,x2).
(a dependency relation)
3x1+2x2+(?1)x3?0Since the three coefficients are nonzero, we could solve for any vector in terms of the other two. It follows that
Span(x1,x2,x3)=Span(x1,x2)=Span(x1,x3)=Span(x2,x3)
On the other hand, no such dependency relationship exists between
x1 and x2. In deed, if there were scalars c1 and
c2, not both 0, such that
(3)
c1x1+c2x2?0
then we could solve for one of the two vectors in terms of the other. However, neither of the two vectors in question is a multiple of the other. Therefore, Span(x1) and Span(
x2) are both proper subspaces of
c1=c2=0Span(x1,x2), and the only way that (3) can hold is if
Observations: (I)
If
v1,v2,?,vn.
span a vector space V and one of these vectors can
be written as a linear combination of the other n-1 vectors, then those n-1 vectors span V.
Chapter 3---Section 3 Linear Independence
(II) Given n vectors
v1,v2,?,vn, it is possible to write one of the
vectors as a linear combination of the other n-1 vectors if and only if there exist scalars
c1,c2,?,cn
not all zero such that
c1v1?c2v2???cnvn?0
Proof of I: Suppose that the vectors
v1,v2,?,vn-1.
vn
can be written as a linear combination of
Proof of II: The key point here is that there at least one nonzero coefficient.
3.2 Definitions
★Definition The vectors
v1,v2,?,vn in a vector space V are said to
be linearly independent（线性独立的） if
c1v1?c2v2???cnvn?0
implies that all the scalars Example:
Definition The vectors
e1,e2,?,en
c1,c2,?,cn
must equal zero.
are linearly independent.
v1,v2,?,vn in a vector space V are said to be
c1,c2,?,cn
linearly dependent （线性相关的）if there exist scalars zero such that c1v1?c2v2???cnvn?0not all .
RnLet
e1,e2,?,en,x be vector in . Then
e1,e2,?,en,x are linearly
Chapter 3---Section 3 Linear Independence
dependent.
If there are nontrivial choices of scalars for which the linear combination
c1v1?c2v??2?cvn
equals the zero vector, then
v1,v2,?,vn
are linearly dependent. If the only way the linear combination
c1v1?c2v2???cnvn can equal the zero vector is for all scalars
are linearly independent.
c1,c2,?,cn
to be 0, then
v1,v2,?,vn3.3 Geometric Interpretation
The linear dependence and independence in Each vector in
R2R2 and
R3.
or
R3 represents a directed line segment
originated at the origin.
Two vector are linearly dependent in
R2 or
R3 if and only if two
2vectors are collinear. Three or more vector in dependent. Three vectors in
R3R must be linearly
are linearly dependent if and only if three
R3vectors are coplanar. Four or more vectors in dependent.
must be linearly
Chapter 3---Section 3 Linear Independence
3.4 Theorems and Examples
In this part, we learn some theorems that tell whether a set of vectors is linearly independent.
Example: (Example 3 on page 138) Which of the following collections of vectors are linearly independent?
(a)
?e1e21,0,2,e31?,1?,TT?1,?1,?0,2,1,1,3,?T?
?1,0,0?T(b) ??1,(c) ??1,(d) ??1,
0?,0?TT?
T?
?4,?1,1?4?,T?2,1,3?,T?
The problem of determining the linear dependency of a collection of vectors in
Rm can be reduced to a problem of solving a linear
homogeneous system.
If the system has only the trivial solution, then the vectors are linearly independent, otherwise, they are linearly dependent, We summarize the this method in the following theorem:
Theorem n vectors
x1,x2,?,xn in
Rm are linearly dependent if the
X=(x1,x2,?,xn)linear system Xc=0 has a nontrivial solution, where Proof:
c1x1+c2x2+??cnxn?0.
? Xc=0.
Chapter 3---Section 3 Linear Independence
Theorem 3.3.1 Let
X=(x1,x2,?,xn)x1,x2,?,xn be n vectors in
Rn and let
. The vectors
x1,x2,?,xn
will be linearly dependent if and
only if X is singular. (the determinant of X is zero)
Proof: Xc=0 has a nontrivial solution if and only X is singular.
Theorem 3.3.2 Let
v1,v2,?,vn be vectors in a vector space V. A vector v
in Span(v1,v2,?,vn) can be written uniquely as a linear combination of
v1,v2,?,vn
if and only if
v1,v2,?,vn are linearly independent.
(A vector v in Span(v1,v2,?,vn) can be written as two different linear combinations of dependent.)
（Note: If---sufficient condition ; Only if--- necessary condition） Proof: Let v? Span(v1,v2,?,vn), then
v??1v1??2v2????nvnv1,v2,?,vn if and only if
v1,v2,?,vn are linearly
Necessity: (contrapositive law for propositions)
Suppose that vector v in Span(v1,v2,?,vn) can be written as two different linear combination of
v1,v2,?,vn, then prove that
v1,v2,?,vn are
linearly dependent. The difference of two different linear combinations gives a dependency relation of Suppose that
v1,v2,?,vnv1,v2,?,vn
are linearly dependent, then there exist two