Linear algebra with explanative examples.pdf

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1. 1.7 Linear Independence
Math 2331 { Linear Algebra
1.7 Linear Independence
Jiwen He
Department of Mathematics, University of Houston
[email protected]
math.uh.edu/jiwenhe/math2331
Jiwen He, University of Houston Math 2331, Linear Algebra 1 / 17

2. 1.7 Linear Independence Denition
1.7 Linear Independence
Linear Independence and Homogeneous System
Linear Independence: Denition
Linear Independence of Matrix Columns
Special Cases
A Set of One Vector
A Set of Two Vectors
A Set Containing the0Vector
A Set Containing Too Many Vectors
Characterization of Linearly Dependent Sets
Theorem: Linear Dependence and Linear Combination
Jiwen He, University of Houston Math 2331, Linear Algebra 2 / 17

3. 1.7 Linear Independence Denition
Linear Independence and Homogeneous System
Example
A homogeneous system such as
2
4
1 23
3 5 9
5 9 3
3
5
2
4
x1
x2
x3
3
5=
2
4
0
0
0
3
5
can be viewed as a vector equation
x1
2
4
1
3
5
3
5+x2
2
4
2
5
9
3
5+x3
2
4
3
9
3
3
5=
2
4
0
0
0
3
5:
The vector equation has the trivial solution (x1= 0,x2= 0,
x3= 0), but is this theonly solution?
Jiwen He, University of Houston Math 2331, Linear Algebra 3 / 17

4. 1.7 Linear Independence Denition
Linear Independence: Denition
Linear Independence
A set of vectorsfv1;v2; : : : ;vpginR
n
is said to belinearly
independentif the vector equation
x1v1+x2v2+ +xpvp=0
has only the trivial solution.
Linear Dpendence
The setfv1;v2; : : : ;vpgis said to belinearly dependentif there
exists weightsc1; : : : ;cp;not all 0, such that
c1v1+c2v2+ +cpvp=0.
"
linear dependence relation
(when weights are not all zero)
Jiwen He, University of Houston Math 2331, Linear Algebra 4 / 17

5. 1.7 Linear Independence Denition
Linear Independence: Example
Example
Letv1=
2
4
1
3
5
3
5,v2=
2
4
2
5
9
3
5,v3=
2
4
3
9
3
3
5.
a. Determine iffv1;v2;v3gis linearly independent.
b. If possible, nd a linear dependence relation amongv1;v2;v3.
Solution: (a)
x1
2
4
1
3
5
3
5+x2
2
4
2
5
9
3
5+x3
2
4
3
9
3
3
5=
2
4
0
0
0
3
5.
Augmented matrix:
2
4
1 23 0
3 5 9 0
5 9 3 0
3
5
2
4
1 2 3 0
01 18 0
01 18 0
3
5
2
4
1 2 3 0
01 18 0
0 0 0 0
3
5
x3is a free variable)there are nontrivial solutions.
Jiwen He, University of Houston Math 2331, Linear Algebra 5 / 17

6. 1.7 Linear Independence Denition
Linear Independence: Example (cont.)
)fv1;v2;v3gis a linearly dependent set
(b) Reduced echelon form:
2
4
1 0 33 0
0 118 0
0 0 0 0
3
5=)
x1=
x2=
x3
Letx3= (any nonzero number).
Thenx1= andx2= .
2
4
1
3
5
3
5+
2
4
2
5
9
3
5+
2
4
3
9
3
3
5=
2
4
0
0
0
3
5
or
v1+v2+v3=0
(one possible linear dependence relation)
Jiwen He, University of Houston Math 2331, Linear Algebra 6 / 17

7. 1.7 Linear Independence Denition
Linear Independence of Matrix Columns
Example (Linear Dependence Relation)
33
2
4
1
3
5
3
5+ 18
2
4
2
5
9
3
5+ 1
2
4
3
9
3
3
5=
2
4
0
0
0
3
5
can be written as the matrix equation:
2
4
1 23
3 5 9
5 9 3
3
5
2
4
33
18
1
3
5=
2
4
0
0
0
3
5. Each linear dependence relation among the columns ofA
corresponds to a nontrivial solution toAx=0.
The columns of matrixAare linearly independent if and only if the
equationAx=0hasonlythe trivial solution.
Jiwen He, University of Houston Math 2331, Linear Algebra 7 / 17

8. 1.7 Linear Independence Denition
Special Cases: 1. A Set of One Vector
Sometimes we can determine linear independence of a set with
minimal eort.
Example (1. A Set of One Vector)
Consider the set containing one nonzero vector:fv1g
The only solution tox1v1= 0 isx1= : Sofv1gis linearly independent whenv16=0:
Jiwen He, University of Houston Math 2331, Linear Algebra 8 / 17

9. 1.7 Linear Independence Denition
Special Cases: 2. A Set of Two Vectors
Example (2. A Set of Two Vectors)
Let
u1=

2
1

,u2=

4
2

;v1=

2
1

;v2=

2
3

:
a. Determine iffu1;u2gis a linearly dependent set or a linearly
independent set.
b. Determine iffv1;v2gis a linearly dependent set or a linearly
independent set.
Solution:(a) Notice thatu2= u1. Therefore
u1+ u2= 0
This means thatfu1;u2gis a linearly set.
Jiwen He, University of Houston Math 2331, Linear Algebra 9 / 17

10. 1.7 Linear Independence Denition
Special Cases: 2. A Set of Two Vectors (cont.)
(b) Suppose
cv1+dv2=0.
Thenv1= v2ifc6= 0. But this is impossible sincev1is
a multiple ofv2which meansc= .
Similarly,v2= v1ifd6= 0.
But this is impossible sincev2is not a multiple ofv1and sod= 0.
This means thatfv1;v2gis a linearly set.
Jiwen He, University of Houston Math 2331, Linear Algebra 10 / 17

11. 1.7 Linear Independence Denition
Special Cases: 2. A Set of Two Vectors (cont.)
A set of two vectors is linearly dependent if at least one vector is a
multiple of the other.
A set of two vectors is linearly independent if and only if neither of
the vectors is a multiple of the other.
linearly linearly
Jiwen He, University of Houston Math 2331, Linear Algebra 11 / 17

12. 1.7 Linear Independence Denition
Special Cases: 3. A Set Containing the0Vector
Theorem
A set of vectorsS=fv1;v2; : : : ;vpginR
n
containing the zero
vector is linearly dependent.
Proof:Renumber the vectors so thatv1=. Then
v1+ v2+ + vp=0
which shows thatSis linearly .
Jiwen He, University of Houston Math 2331, Linear Algebra 12 / 17

13. 1.7 Linear Independence Denition
Special Cases: 4. A Set Containing Too Many Vectors
Theorem
If a set contains more vectors than there are entries in each vector,
then the set is linearly dependent. I.e. any setfv1;v2; : : : ;vpgin
R
n
is linearly dependent ifp>n.
Outline of Proof:
A=

v1v2 vp

isnp
Supposep>n:
=)Ax=0has more variables than equations
=)Ax=0has nontrivial solutions
=)columns ofAare linearly dependent
Jiwen He, University of Houston Math 2331, Linear Algebra 13 / 17

14. 1.7 Linear Independence Denition
Special Cases: Examples
Examples
With the least amount of work possible, decide which of the
following sets of vectors are linearly independent and give a reason
for each answer.
a.
8
<
:
2
4
3
2
1
3
5;
2
4
9
6
4
3
5
9
=
;
b. Columns of
2
6
6
4
1 2 3 4 5
6 7 8 9 0
9 8 7 6 5
4 3 2 1 8
3
7
7
5
Jiwen He, University of Houston Math 2331, Linear Algebra 14 / 17

15. 1.7 Linear Independence Denition
Special Cases: Examples (cont.)
Examples (cont.)
c.
8
<
:
2
4
3
2
1
3
5;
2
4
9
6
3
3
5;
2
4
0
0
0
3
5
9
=
;
d.
8
>
>
<
>
>
:
2
6
6
4
8
2
1
4
3
7
7
5
9
>
>
=
>
>
;
Jiwen He, University of Houston Math 2331, Linear Algebra 15 / 17

16. 1.7 Linear Independence Denition
Characterization of Linearly Dependent Sets
Example
Consider the set of vectorsfv1;v2;v3;v4ginR
3
in the following
diagram. Is the set linearly dependent? Explain
Jiwen He, University of Houston Math 2331, Linear Algebra 16 / 17

17. 1.7 Linear Independence Denition
Characterization of Linearly Dependent Sets
Theorem
An indexed setS=fv1;v2; : : : ;vpgof two or more vectors is
linearly dependent if and only if at least one of the vectors inSis a
linear combination of the others. In fact, ifSis linearly
dependent, andv16=0, then some vectorvj(j2) is a linear
combination of the preceding vectorsv1; : : : ;vj1.
Jiwen He, University of Houston Math 2331, Linear Algebra 17 / 17