MATH 115A  Assignment One  Solutions of Select NonGraded
Problems
Paul Skoufranis
October 3, 2011
§
1.2 Question 13)
Let
V
denote the set of ordered pairs of real numbers.
If
(
a
1
, a
2
)
and
(
b
1
, b
2
)
are
elements of
V
and
c
∈
R
, define
(
a
1
, a
2
) + (
b
1
, b
2
) = (
a
1
+
b
1
, a
2
b
2
)
and
c
·
(
a
1
, a
2
) = (
ca
1
, a
2
)
Is
V
a vector space over
R
with these operations? Justify your answer.
Solution
:
The set
V
with the above operations is not a vector space.
We will provide two ways to
see this:
1. Property (VS 8) in the definition of a vector space fails for
V
with these operations. To see this, we
notice that if
~v
= (1
,
2), then
(1 + 1)
·
~v
= 2
·
~v
= (2
,
2)
yet
1
·
~v
+ 1
·
~v
= 1
·
(1
,
2) + 1
·
(1
,
2) = (1
,
2) + (1
,
2) = (2
,
4)
so (1 + 1)
·
~v
6
= 1
·
~v
+ 1
·
~v
. Hence
V
is not a vector space.
2. Suppose
V
is a vector space. Then
V
must have a unique zero vector. Since
(
a, b
) + (0
,
0) = (
a
+ 0
, b
(0)) = (
a,
0)
,
~v
+ (0
,
0) =
~v
for all
~v
∈
V
. Hence (0
,
0) must be the zero vector of
V
. However, we notice that
(1
,
0) + (

1
,
0) = (0
,
0) = (1
,
0) + (

1
,
1)
.
Therefore (

1
,
0) and (

1
,
1) are distinct additive inverse of (1
,
0).
Since every element of a vector
space has a unique additive inverse, we have a contradiction. Therefore
V
is not a vector space.
Thus we have demonstrated that
V
is not a vector space in two different ways.
§
1.2 Question 17)
Let
V
=
{
(
a
1
, a
2
)

a
1
, a
2
∈
F
}
, where
F
is a field.
Define addition of elements
of
V
coordinate wise, and for
c
∈
F
and
(
a
1
, a
2
)
∈
V
, define
c
·
(
a
1
, a
2
) = (
ca
1
,
0)
Is
V
a vector space over
F
with these operations? Justify your answer.
Solution
:
The set
V
with the above operations is not a vector space.
The easiest way to see this is
to note (using the know properties of fields) that 1
·
(1
,
1) = (1
,
0). Since 1
6
= 0 in any field, (1
,
1)
6
= (1
,
0) so
1
·
(1
,
1)
6
= (1
,
1). Hence property (VS 5) in the definition of a vector space fails for
V
with these operations.
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§
1.2 Question 21)
Let
V
and
W
be vector spaces over a field
F
with respect to the operations
+
V
,
+
W
,
·
V
, and
·
W
. Let
Z
=
{
(
~v, ~w
)

~v
∈
V, ~w
∈
W
}
Prove that
Z
is a vector space with the operations
(
~v
1
, ~w
1
) + (
~v
2
, ~w
2
) = (
~v
1
+
V
~v
2
, ~w
1
+
W
~v
2
)
and
c
·
(
~v, ~w
) = (
c
·
V
~v, c
·
W
~w
)
for all
~v,~v
1
,~v
2
∈
V
,
~w, ~w
1
, ~w
2
∈
W
, and
c
∈
F
.
Proof
: To show that
Z
is a vector space over
F
with the above operations, we will show that
Z
satis
fies the definition of a vector space.
It is clear that the operations + and
·
are welldefined (that is, if
~v,~v
1
,~v
2
∈
V
,
~w, ~w
1
, ~w
2
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 Fall '08
 Kong,L
 Math, Real Numbers, Addition, Vector Space, scalar multiplication, additive inverse

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