线性代数 英文讲义 下载本文

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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