EE 103, Spring ’09, Prof SEJ: HW 4 Sol.
Page
1
of
7
Applied Numerical Computing
Instructor: Prof. S. E. Jacobsen
HW 4 Solution
Students: Distributed HW solutions are a component of the course and should be fully
understood.
Prob 1:
1
n
a x
Ax
a x
. Then,
1
1,
,
1,
,
1
1,
,
max
,
,
max
,
,
max
,
,
max{
,
,
}max
,
,
.
n
j
j
n j
j
j
j
j
j
n j
j
j
j
n
j
n j
j
j
Ax
a x
a x
a
x
a
x
a
x
a
x
x
x
a
a
x
Ax
Prob 2:
(a)
x
is the unique solution of
Ax
b
.
If the columns of
A
are linearly dependent, there exists
a vector
ˆ
0
x
so that
ˆ
0
Ax
.
Consider the vector
ˆ
x
x
x
x
.
Then
ˆ
Ax
Ax
Ax
b
and the solution
x
is not unique.
(b)
Consider the equation
Lx
b
.
By forward substitution, it’s clear that the solution is unique.
By part (a), the columns of
L
are linearly independent and
L
is nonsingular. Or, consider the
equation
0
Lx
; by forward substitution, it’s clear that the unique solution is
0
x
.

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