2
Theorem 2
.
3
If
[
A

0
]
is a homogeneous system of
m
linear
equations with
n
variables, where
m
<
n
, then the system has
infinitely many solutions.
Proof.
After applying GaussJordan Elimination to
[
A

0
]
, we get
[
B

0
]
in
reduced row echelon form. We make two observations on
[
B

0
]
:
(1)
[
B

0
]
is consistent:
If
B
has an allzeros row, then that row
corresponds to the equation 0
=
0 because it is homogenous. So, it
has at least one solution.
(2)
It has at least one free variable:
Since
B
has m equations, it can at
most have
m
nonzero rows. Hence, it has at least
n

m
>
0 free
variables.
§
1.1 and
§
1.2
1.26
Homogeneous Systems
1
Definition
A system of linear equations is called
homogeneous
of the constant term in each equation is zero. Otherwise, it is
called
non

homogeneous
. If the
[
A

b
]
is homogeneous, then
b
=
0
.
2
Theorem 2
.
3
If
[
A

0
]
is a homogeneous system of
m
linear
equations with
n
variables, where
m
<
n
, then the system has
infinitely many solutions.
Proof.
After applying GaussJordan Elimination to
[
A

0
]
, we get
[
B

0
]
in
reduced row echelon form. We make two observations on
[
B

0
]
:
(1)
[
B

0
]
is consistent:
If
B
has an allzeros row, then that row
corresponds to the equation 0
=
0 because it is homogenous. So, it
has at least one solution.
(2)
It has at least one free variable:
Since
B
has m equations, it can at
most have
m
nonzero rows. Hence, it has at least
n

m
>
0 free
variables.
(
1
)
and
(
2
)
together imply that the system has infinitely many
solutions. (each free variable introduces a new variable in the solution
set)
§
1.1 and
§
1.2
1.27
What if it is nonhomogeneous?
The last row in the following SLEs gives 0
=
1. This is not possible. So,
it has no solution.
1
0
1
0
1
2
0
0
1
§
1.1 and
§
1.2
1.27
What if it is nonhomogeneous?
The last row in the following SLEs gives 0
=
1. This is not possible. So,
it has no solution.
1
0
1
0
1
2
0
0
1
Remark
1
:
An SLEs has NO solution (inconsistent), if its augmented
matrix
[
A

b
]
is row equivalent to some
[
B

c
]
such that one of the rows
of
[
B

c
]
has the form
£
0
0
...
0
0
d
6=
0
/
§
1.1 and
§
1.2
1.27
What if it is nonhomogeneous?
The last row in the following SLEs gives 0
=
1. This is not possible. So,
it has no solution.
1
0
1
0
1
2
0
0
1
Remark
1
:
An SLEs has NO solution (inconsistent), if its augmented
matrix
[
A

b
]
is row equivalent to some
[
B

c
]
such that one of the rows
of
[
B

c
]
has the form
£
0
0
...
0
0
d
6=
0
/
Remark
2
:
It is enough to apply elementary row operations and
GaussJordan Elimination to
[
A

b
]
. The process will itself tell you
whether the system is consistent or inconsistent.
§
1.1 and
§
1.2
1.28
CAUTION
THE QUIZ WILL COVER EVERYTHING IN 1.1 AND
1.2
EXCLUDING REDUCED ROW ECHELON
FORMS
AS WE HAVE NOT COVERED IT YET.