Download Linear Independence and Linear Dependence

Linear Independence and Linear Dependence
De…nition 1 An indexed set of vectors fv1 ; v2 ; : : : ; vn g in Rm is said to be
linearly independent if the vector equation
x1 v1 + x2 v2 +
+ xn vn = 0m
has only the trivial solution (x1 = x2 =
= xn = 0).
If the above vector equation has non–trivial solutions, then the set of
vectors fv1 ; v2 ; : : : ; vn g is said to be linearly dependent and any equation
of the form
c1 v1 + c2 v2 +
+ cn vn = 0m
with not all of the numbers c1 ,c2 ,. . . ,cn equal to zero is called a linear dependence relation for the set fv1 ; v2 ; : : : ; vn g.
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Example 2 Let
2
2
2
3
3
3
0
0
3
v1 = 4 0 5 , v2 = 4 5 5 , and v3 = 4 4 5 .
2
8
1
1. Determine whether the set fv1 ; v2 ; v3 g is linearly independent or linearly dependent.
2. If the set fv1 ; v2 ; v3 g is linearly dependent, then write a linear dependence relation for this set.
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Example 3 Let
2
2
2
3
3
3
0
3
3
v1 = 4 0 5 , v2 = 4 4 5 , and v3 = 4 4 5 .
2
7
1
1. Determine whether the set fv1 ; v2 ; v3 g is linearly independent or linearly dependent.
2. If the set fv1 ; v2 ; v3 g is linearly dependent, then write a linear dependence relation for this set.
3
Remark 4 Suppose that A is an m n matrix. Then the set of vectors
(in Rm ) that make up the columns of A form a linearly independent set if
and only if the homogeneous matrix equation Ax = 0m has only the trivial
solution.
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Determining Linear Independence/Dependence For Sets
of One or Two Vectors
One Vector
Suppose that we have a single vector v1 2 Rm . We can easily tell
whether the set fv1 g is linearly independent or linearly dependent:
If v1 = 0m , then we have the linear dependence relation
1 v1 = 0m
which shows that the set fv1 g is linearly dependent.
However, if v1 6= 0m , then the vector equation
x1 v1 = 0m
has only the trivial solution, which means that the set fv1 g is linearly independent.
Summary 5
1. If v1 = 0m , then the set fv1 g is linearly dependent.
2. If v1 6= 0m , then the set fv1 g is linearly independent.
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Two Vectors
Suppose that we have two vectors v1 and v2 2 Rm . We can easily tell
whether the set fv1 ; v2 g is linearly independent or linearly dependent:
Suppose that v2 is a scalar multiple of v1 . Then v2 = cv1 for some
scalar c. This means that we have the linear dependence relation
c v1 + 1 v2 = 0m
and hence that the set fv1 ; v2 g is linearly dependent. Likewise, if v1 is a
scalar multiple of v2 , then the set fv1 ; v2 g is linearly dependent.
Conversely, suppose that the set fv1 ; v2 g is linearly dependent. Then
we can write some linear dependence relation
c1 v1 + c2 v2 = 0m
where either c1 6= 0 or c2 6= 0. If c1 6= 0, then
c2
c1
v2 =
v1
which means that v2 is a scalar multiple of v1 . If c2 6= 0, then
c1
c2
v1 =
v2
which means that v1 is a scalar multiple of v2 . Thus if the set fv1 ; v2 g is
linearly dependent, then either v2 must be a scalar multiple of v1 or v1 must
be a scalar multiple of v2 .
Summary 6
1. If v2 is a scalar multiple of v1 or if v1 is a scalar multiple
of v2 , then the set fv1 ; v2 g is linearly dependent.
2. If neither of the vectors v1 and v2 is a scalar multiple of the other one,
then the set fv1 ; v2 g is linearly independent.
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Example 7 Let
v1 =
1
4
and v2 =
2
8
.
Is the set of vectors fv1 ; v2 g a linearly independent set or a linearly dependent
set? If it is linearly dependent, then write a linear dependence relation for
this set.
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Determining Linear Independence/Dependence For Sets
Containing the Zero Vector
If we have a set of vectors fv1 ; v2 ; : : : ; vn g in Rm and one of the vectors
in this set, say vj , is the zero vector (0m ), then this set is linearly dependent
because we have the linear dependence relation
0 v1 + 0 v2 +
+ 1 vj +
Example 8 Suppose that
2
3
2
3
2
0
3
1
6 6 7
6 0
6 6 7
6
7
6
7
v1 = 6
4 1 5 , v2 = 4 1 5 , v3 = 4 0
0
14
12
+ 0 vn = 0m .
3
0
6
7
7
7 , and v4 = 6 5 7 .
5
4 0 5
1
3
2
Write a linear dependence relation for the set of vectors fv1 ; v2 ; v3 ; v4 g.
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Determining Linear Independence/Dependence For Sets
Containing More Vectors Than There Are Entries in
Each Vector
If we have a set of vectors fv1 ; v2 ; : : : ; vn g in Rm and n > m (in other
words, there are more vectors in this set than there are entries in each vector),
then the set fv1 ; v2 ; : : : ; vn g is linearly dependent.
Here is why: If we let A be the m n matrix whose columns are the
vectors v1 , v2 ,. . . ,vn , then A has m rows and hence rref(A) can have at most
m row–leading 1s. Since A has n columns and n > m, then not every column
of rref(A) can have a row–leading 1. This means that the homogeneous
matrix equation Ax = 0m has non–trivial solutions (because there must be
some free variables), and hence that the set of vectors fv1 ; v2 ; : : : ; vn g is
linearly dependent.
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Example 9 Let
v1 =
2
0
, v2 =
3
5
, and v3 =
8
6
.
Explain (without doing any computations) how you know that the set of vectors fv1 ; v2 ; v3 g is linearly dependent. Then write a linear dependence relation for this set. (This latter part does require computation.)
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A General Characterization of Linear Dependence
Theorem 10 Suppose that fv1 ; v2 ; : : : ; vn g is a set of two or more vectors
in Rm . This set of vectors is linearly dependent if and only if at least one of
the vectors in this set is a linear combination of the other vectors in the set.
Proof. Suppose that fv1 ; v2 ; : : : ; vn g is a set of two or more vectors in
R and suppose that this set of vectors is linearly dependent. Then we have
a linear dependence relation
m
+ cn vn = 0m .
c1 v1 + c2 v2 +
Not all of the numbers c1 , c2 , : : : ; cn are zero. In particular, there is some
index j such that cj 6= 0. This means that
vj =
c1
cj
v1 +
+
cj 1
cj
vj
1
cj+1
cj
+
vj+1 +
+
cn
cj
vn
showing that vj is a linear combination of the other vectors in the set.
Conversely, suppose that there is some index j such that vj is a linear
combination of the other vectors in the set. Then
vj = c1 v2 + c2 v2 +
+ cj 1 vj
1
+ cj+1 vj+1 +
+ cn vn
This means that
c1 v2 + c2 v2 +
+ cj 1 vj
1
+ ( 1) vj + cj+1 vj+1 +
+ cn vn = 0m
and hence that the set fv1 ; v2 ; : : : ; vn g is linearly dependent.
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Example 11 Let
2
2
2
3
3
1
1
v1 = 4 4 5 , v2 = 4 2 5 , v3 = 4
0
3
2
3
3
1
1
4 5 , v4 = 4 4 5 .
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3
Explain why the set of vectors fv1 ; v2 ; v3 ; v4 g is linearly dependent and
…nd an index j 2 f1; 2; 3; 4g such that vj is a linear combination of the other
three vectors in the set.
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Some Words of Caution
1. We know that if a set of vectors contains more vectors than there
are entries in each vector, then this set must be linearly dependent.
However, there are certainly linearly dependent sets of vectors that do
not contain more vectors than there are entries in each vector.
An example of a set of vectors that is linearly dependent but does
not contain more vectors than there are entries in each vectors is:
2. We know that if a set of vectors is linearly dependent, then it must be
true that at least one vector in the set is a linear combination of the
other vectors in the set. However, it need not be true that all of the
vectors in the set are linear combinations of the other vectors in the
set.
An example of a set of vectors that is linearly dependent but
which contains a vector that is not a linear combination of the other vectors
is:
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