Math 307: Problems for section 2.1
October 4, 2009
1.
Are the vectors
1
2
1
2
1
,
1
0

2
1
1
,
1

1
3

2
0
,
0
0

2
0
1
,
0
4

9
7
3
linearly independent? You may use MAT
LAB/Octave to perform calculations, but explain your answer.
2.
Which of the following sets are subspaces of the vector space
V
? Why, or why not?
(a) The set
S
=
{
(
b
1
, b
2
, b
3
) :
b
1
= 0;
b
2
, b
3
∈
R
}
. (
V
=
R
3
)
(b) The set
S
=
{
(
b
1
, b
2
, b
3
) :
b
1
b
2
= 0
, b
3
∈
R
}
.
This is union of the plane
b
1
= 0
and the
plane
b
2
= 0
. (
V
=
R
3
)
(c) All infinite sequences
(
x
1
, x
2
, . . .
)
, with
x
i
∈
R
and
x
j
= 0
from some fixed point
onwards. (
V
=
R
∞
)
(d) All nonincreasing sequences
(
x
1
, x
2
, . . .
)
, with
x
i
∈
R
and
x
j
+1
≤
x
j
for each
j
.
(
V
=
R
∞
)
(e) The set of all polynomial functions,
p
(
x
)
, where
p
(
x
) = 0
or
p
(
x
)
has degree
n
for some
fixed
n
≥
1
. (
V
is the vector space of all polynomials.)
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Chapter 7 / Exercise 24
Mathematical Applications for the Management, Life, and Social Sciences
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