CO350
Assignment 1 Solutions
Exercise 1:
(linear algebra review.) Let
A
∈
R
m
×
n
and
b
∈
R
m
.
(a)
If the system
Ax
=
b
has a solution and the columns of
A
are not linearly indepen
dent, then
Ax
=
b
has inﬁnitely many solutions.
Proof.
Suppose that
x
0
∈
R
n
satisﬁes
Ax
0
=
b
. If the columns of
A
are not linearly
independent, then there is a nonzero vector
x
1
∈
R
n
such that
Ax
1
= 0. Now, for each
a
∈
R
, we have
A
(
x
0
+
ax
1
) =
Ax
0
+
aAx
1
=
b
. Hence
Ax
=
b
has inﬁnitely many
solutions.
(b)
Show that, if the system
Ax
=
b
has a solution and the columns of
A
are linearly
independent, then
Ax
=
b
has a unique solution.
Proof.
We will prove the equivalent statement that, if
Ax
=
b
has two distinct solutions,
then the columns of
A
are not linearly independent. Suppose that
x
1
∈
R
n
and
x
2
∈
R
n
are distinct solutions of
Ax
=
b
. Then
x
=
x
2

x
1
is a nonzero vector and
Ax
=
Ax
2

Ax
1
= 0. Hence, the columns of
A
are not linearly independent.
Exercise 2:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '07
 S.Furino,B.Guenin
 Vector Space, Optimization, Kilogram

Click to edit the document details