lay linalg5!01!07
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1.7
2012 Pearson Education, Inc.
Linear Equations
in Linear Algebra
LINEAR INDEPENDENCE
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Slide 1.7- 2 2012 Pearson Education, Inc.
LINEAR INDEPENDENCE
Definition:An indexed set of vectors {v1, , vp} in
is said to be linearly independentif the vector
equation
has only the trivial solution. The set {v1, , vp} is
said to be linearly dependentif there exist weights
c1, , cp, not all zero, such that----(1)
n
1 1 2 2
v v ... v 0p p
x x x
1 1 2 2v v ... v 0
p pc c c
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Slide 1.7- 3 2012 Pearson Education, Inc.
LINEAR INDEPENDENCE
Equation (1) is called a linear dependence relation
among v1, , vp when the weights are not all zero.
An indexed set is linearly dependent if and only if it
is not linearly independent.
Example 1:Let , , and .1
1
v 2
3
2
4
v 5
6
3
2
v 1
0
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Slide 1.7- 4 2012 Pearson Education, Inc.
a. Determine if the set {v1, v2, v3} is linearlyindependent.
b. If possible, find a linear dependence relation
among v1, v2, and v3.
Solution:We must determine if there is a nontrivial
solution of the following equation.
1 2 3
1 4 2 0
2 5 1 0
3 6 0 0
x x x
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Slide 1.7- 5 2012 Pearson Education, Inc.
LINEAR INDEPENDENCE
Row operations on the associated augmented matrix
show that
.
x1andx2are basic variables, andx3is free.
Each nonzero value ofx3determines a nontrivialsolution of (1).
Hence, v1, v2, v3are linearly dependent.
1 4 2 0 1 4 2 0
2 5 1 0 0 3 3 03 6 0 0 0 0 0 0
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Slide 1.7- 6 2012 Pearson Education, Inc.
LINEAR INDEPENDENCE
b. To find a linear dependence relation among v1,
v2, and v3, row reduce the augmented matrix
and write the new system:
Thus, , , andx3is free. Choose any nonzero value forx3say, .
Then and .
1 0 2 0
0 1 1 0
0 0 0 0
1 3
2 3
2 0
0
0 0
x x
x x
1 32x x 2 3x x 3
5x
1 10x
2 5x
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7/20Slide 1.7- 7 2012 Pearson Education, Inc.
LINEAR INDEPENDENCE
Substitute these values into equation (1) and obtain
the equation below.
This is one (out of infinitely many) possible linear
dependence relations among v1, v2, and v3.
1 2 3
10v 5v 5v 0
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LINEAR INDEPENDENCE OF MATRIX COLUMNS
Suppose that we begin with a matrix
instead of a set of vectors.
The matrix equation can be written as
.
Each linear dependence relation among the columns ofA corresponds to a nontrivial solution of .
Thus, the columns of matrixAare linearly independent ifand only if the equation has only the trivial
solution.
1a a
n
A
x 0A
1 1 2 2a a ... a 0n nx x x
x 0A
x 0A
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9/20Slide 1.7- 9 2012 Pearson Education, Inc.
SETS OF ONE OR TWO VECTORS
A set containing only one vectorsay, vis linearly
independent if and only if vis not the zero vector.
This is because the vector equation has onlythe trivial solution when .
The zero vector is linearly dependent becausehas many nontrivial solutions.
1v 0x v 0
10 0x
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SETS OF ONE OR TWO VECTORS
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.
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11/20Slide 1.7- 11 2012 Pearson Education, Inc.
SETS OF TWO OR MORE VECTORS
Theorem 7: Characterization of Linearly Dependent
Sets
An indexed set of two or more
vectors is linearly dependent if and only if at least oneof the vectors in Sis a linear combination of the
others.
In fact, if Sis linearly dependent and , then
some vj(with ) is a linear combination of the
preceding vectors, v1, , .
1{v ,...,v }
pS
1
v 01j
1v
j
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SETS OF TWO OR MORE VECTORS
Proof:If some vjin Sequals a linear combination ofthe other vectors, then vjcan be subtracted from both
sides of the equation, producing a linear dependence
relation with a nonzero weight on vj.
[For instance, if , then
.]
Thus Sis linearly dependent.
Conversely, suppose Sis linearly dependent.
If v1is zero, then it is a (trivial) linear combination of
the other vectors in S.
( 1)
1 2 2 3 3v v vc c
1 2 2 3 3 40 ( 1)v v v 0v ... 0v
pc c
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13/20Slide 1.7- 13 2012 Pearson Education, Inc.
SETS OF TWO OR MORE VECTORS
Otherwise, , and there exist weights c1, , cp, not
all zero, such that
.
Letjbe the largest subscript for which .
If , then , which is impossible because
.
1v 0
1 1 2 2v v ... v 0
p pc c c
0j
c
1j 1 1v 0c
1v 0
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Slide 1.7- 14 2012 Pearson Education, Inc.
SETS OF TWO OR MORE VECTORS
So , and1j
1 1 1v ... v 0v 0v ... 0v 0
j j j j pc c
1 1 1 1v v ... vj j j jc c c
11
1 1v v ... v .
j
j j
j j
cc
c c
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Slide 1.7- 15 2012 Pearson Education, Inc.
SETS OF TWO OR MORE VECTORS
Theorem 7 does notsay that everyvector in a linearly
dependent set is a linear combination of the precedingvectors.
A vector in a linearly dependent set may fail to be alinear combination of the other vectors.
Example 2:Let and . Describe the
set spanned by uand v, and explain why a vector wisin Span {u, v} if and only if {u, v, w} is linearlydependent.
3
u 1
0
1
v 6
0
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Slide 1.7- 16 2012 Pearson Education, Inc.
SETS OF TWO OR MORE VECTORS
Solution:The vectors uand vare linearly
independent because neither vector is a multiple of
the other, and so they span a plane in .
Span {u, v} is thex1x2-plane (with ).
If wis a linear combination of uand v, then {u, v, w}is linearly dependent, by Theorem 7.
Conversely, suppose that {u, v, w} is linearly
dependent. By theorem 7, some vector in {u, v, w} is a linear
combination of the preceding vectors (since ).
That vector must be w, since vis not a multiple of u.
3
3 0x
u 0
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Slide 1.7- 17 2012 Pearson Education, Inc.
SETS OF TWO OR MORE VECTORS
So wis in Span {u, v}. See the figures given below.
Example 2 generalizes to any set {u, v, w} in with
uand vlinearly independent.
The set {u, v, w} will be linearly dependent if and
only if wis in the plane spanned by uand v.
3
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Slide 1.7- 18 2012 Pearson Education, Inc.
SETS OF TWO OR MORE VECTORS
Theorem 8:If a set contains more vectors than thereare entries in each vector, then the set is linearly
dependent. That is, any set {v1, , vp} in is
linearly dependent if .
Proof: Let .
ThenAis , and the equation
corresponds to a system of nequations inp
unknowns. If , there are more variables than equations, so
there must be a free variable.
n
p n
1v v
pA
n p x 0A
p n
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Slide 1.7- 19 2012 Pearson Education, Inc.
SETS OF TWO OR MORE VECTORS
Hence has a nontrivial solution, and thecolumns of A are linearly dependent.
See the figure below for a matrix version of this
theorem.
Theorem 8 says nothing about the case in which the
number of vectors in the set does notexceed the
number of entries in each vector.
x 0A
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Slide 1.7- 20 2012 Pearson Education, Inc.
SETS OF TWO OR MORE VECTORS
Theorem 9:If a set in contains
the zero vector, then the set is linearly dependent.
Proof:By renumbering the vectors, we may suppose
.
Then the equation showsthat Sin linearly dependent.
1{v ,...,v }
pS
n
1v 0
1 21v 0v ... 0v 0p
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