MA 265 LECTURE NOTES: WEDNESDAY, MARCH 19
Homogeneous Systems
Relationship with Nonhomogeneous Systems.
In general, consider a nonhomogeneous system
A
x
=
b
.
Recall that the set of solutions does not form a linear space:
x
∈
R
n
A
x
=
b
.
Say that
x
p
is a particular solution, and let
x
be
any
solution. Denote
x
h
=
x

x
p
. Then we find that
A
x
h
= (
A
x
)

(
A
x
p
) =
b

b
=
0
.
Hence
x
h
is a solution to the homogeneous equation. We summarize this as follows:
Any nonhomogeneous system
A
x
=
b
has the general solution
x
=
x
p
+
x
h
where
x
p
is
one
solution to
the nonhomogeneous system and
x
h
is the
general
solution to the homogeneous system. In other words,
x
∈
R
n
A
x
=
b
=
x
p
+
x
h
x
h
∈
W
=
x
p
+
W
where
W
=
x
∈
R
n
A
x
=
0
.
Here
W
is a linear space; it is the null space of
A
. Similarly,
x
p
+
W
is the
translate
of the linear space
W
.
Example.
Consider the following linear system:
x
+
2
y

3
z
=
2
2
x
+
4
y

6
z
=
4
3
x
+
6
y

9
z
=
6
We can express this in the form
A
x
=
b
in terms of the matrices
A
=
1
2

3
2
4

6
3
6

9
,
x
=
x
y
z
,
and
b
=
2
4
6
.
The augmented matrix for this system is
A
b
=
1
2

3
2
2
4

6
4
3
6

9
6
Performing GaussJordan Reduction gives the reduced row echelon form
1
2

3
2
0
0
0
0
0
0
0
0
=
⇒
x
+ 2
y

3
z
= 2
We see that
y
=
r
and
z
=
s
are arbitrary variables. Hence the general solution is
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 Spring '10
 ...
 Linear Algebra, Space, Row echelon form, row space

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