CAAM 336
·
DIFFERENTIAL EQUATIONS
Problem Set 3
Posted Wednesday 24 January 2007. Due Wednesday 31 January 2007 in class.
1. [20 points]
Consider the following sets. Demonstrate whether or not each is a vector space
(with addition and scalar multiplication defined in the obvious way).
(a)
{
x
∈
R
2
:
x
2
2
=
x
1
}
(b)
{
x
∈
R
3
: 4
x
1
+
x
2
+ 2
x
3
= 0
}
(c)
{
f
∈
C
[0
,
1] :
f
(0)
≥
0
}
(d)
{
f
∈
C
[0
,
1] : max
x
∈
[0
,
1]
f
(
x
)
≤
1
}
(e)
{
f
∈
C
2
[0
,
1] :
d
2
f/dx
2
= 0 for all
x
∈
[0
,
1]
}
2. [24 points]
Recall that a function
f
:
V
→
W
that maps a vector space
V
to a vector space
W
is a
linear operator
provided (1)
f
(
u
+
v
) =
f
(
u
) +
f
(
v
) for all
u, v
in
V
, and (2)
f
(
αv
) =
αf
(
v
) for all
α
∈
R and
v
∈
V
.
Demonstrate whether each of the following functions is a linear operator.
(Show that both properties hold, or give an example showing that one of the properties must fail.)
(a)
f
: R
n
→
R
m
,
f
(
u
) =
Au
for a fixed matrix
A
∈
R
m
×
n
.
(b)
f
: R
n
→
R
m
,
f
(
u
) =
Au
+
b
for a fixed matrix
A
∈
R
m
×
n
and fixed vector
b
∈
R
m
.
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 Fall '09
 Tompson
 Vector Space, matrix equation

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