Homework assignment 3
pp. 3940
Exercise 1.
Which of the following sets
S
of vectors
α
= (
a
1
,...,a
n
)
∈
R
n
are
subspaces of
R
n
(
n
≥
3
)?
(a) all
α
such that
a
1
≥
0;
No. Take
α
= (1
,
0
,...,
0)
∈
S
,
then
(

α
) = (

1
,
0
,...,n
)
/
∈
S.
(b) all
α
such that
a
1
+ 3
a
2
=
a
3
;
Yes. If
α
= (
a
1
,...,a
n
)
,β
= (
b
1
,...,b
n
)
∈
S
,
then
α
+
β,λα
∈
S
, since
(
a
1
+
b
1
) + 3(
a
2
+
b
2
) =
= (
a
1
+ 3
a
2
) + (
b
1
+ 3
b
2
) =
a
3
+
b
3
and
λa
1
+ 3
λa
2
=
λ
(
a
1
+ 3
a
3
) =
λa
2
.
(c) all
α
such that
a
2
=
a
2
1
;
No. Take
α
= (1
,
1
,
0
,...,
0)
∈
S
,
then
2
α
= (2
,
2
,
0
,...,
0)
/
∈
S
.
(d) all
α
such that
a
1
a
2
= 0;
No. Take
α
= (0
,
1
,
0
,...,
0)
,β
= (1
,
0
,...,
0)
∈
S
,
then
α
+
β
= (1
,
1
,
0
,...,
0)
/
∈
S
.
(e) all
α
such that
a
2
is rational. No. Take
α
= (0
,
1
,
0
,...,
0)
,
then
√
2
α /
∈
S
.
Exercise 2.
Let
V
be the (real) vector space of all functions
f
from
R
into
R
. Which of the following sets of functions are subspaces of
V
?
(a) all
f
such that
f
(
x
2
) =
f
(
x
)
2
;
No. Take a constant function
f
(
x
) = 1
for all
x
.
Then
f
∈
S
, but
2
f
∈
S
.
(b) all
f
such that
f
(0) =
f
(1);
Yes. If
f,g
∈
S
, then
f
+
g,λg
∈
S
, since
(
f
+
g
)(0) =
f
(0) +
g
(0) =
f
(1) +
g
(1) =
(
f
+
g
)(1)
,
and
λf
(0) =
λf
(1)
.
(c) all
f
such that
No. Take a function
f
such that
f
(3) = 1
and
f
(3) = 1 +
f
(

5);
f
(
x
) = 0
for all
x
6
= 3
. Then
f
∈
S
, but
2
f /
∈
S
.
(d) all