Unit 17
The Theory of Linear Systems
Theorem 18.1.
Every system of linear equations
Ax
=
b
has either
•
no solution,
or
•
exactly one solution,
or
•
or infinitely many solutions (i.e. parametric family of solutions)
Proof:
We have seen examples of each kind, so we need only show that
there are no more possibilities.
Suppose
y
and
z
are two distinct solutions
i.e.
Ay
=
b
and
Az
=
b
with
z
=
y
for
t
∈
let
x
= (1
−
t
)
y
+
t
(
z
)
(there are infinitely many such
x
’s since
t
is arbitrary).
Then
Ax
=
A
((1
−
t
)
y
+
tz
)
=
A
(
y
+
t
(
z
−
y
))
=
Ay
+
t
(
Az
−
Ay
)
=
b
+
t
(
b
−
b
)
=
b
+ 0 =
b
1
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Therefore, if we have more than one solution, we have infinitely many
solutions.
Definition 18.2.
The linear system
Ax
=
b
is said to be
homogeneous
if
b
= 0. If
b
= 0 then the system is said to be
nonhomogeneous
.
Example 1.
Identify the following systems as homogeneous or nonhomogeneous.
(a)
1
−
1
2
0
0
1
2
0
6
3
2
7
x
y
z
=
0
0
0
homogeneous system.
(b)
3
x
1
+
2
x
2
+
x
3
=
0
3
x
2
−
x
3
=
0
homogeneous system.
2
x
1
+
x
3
=
0
(c)
x
1
−
2
x
2
+
3
x
3
=
0
x
1
−
2
x
3
=
0
nonhomogeneous system.
5
x
2
+
x
3
=
1
Definition 18.3.
The zero vector 0 is a solution to every homogeneous sys
tem
Ax
= 0, since
A
0 = 0. The solution
x
= 0 is called the
trivial solution
,
any other solution is called a
nontrivial solution
.
Theorem 18.4.
Every homogeneous linear system
Ax
= 0
has either ex
actly one solution or infinitely many.
Proof:
We showed that for any linear system there were only 3 options;
no solution, one solution, or infinitely many solutions, and any homogeneous
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 Spring '10
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 Calculus, Linear Equations, Equations, Linear Systems, infinitely many solutions

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