1
1. a)
A field is a set
F
equipped with two binary operations “addition” and “multiplication”
satisfy the following properties:
addition:
AA1:
a
+ (
b
+
c
) = (
a
+
b
) +
c
AA2:
a
+
b
=
b
+
a
AA3:
0, s.t.
a
+ 0 =
a
AA4:
−
a
, s.t.
a
+ (−
a
) = 0
multiplication:
SS1:
a
(
b
c
) = (
a
b
)
c
SS2:
a
b
=
b
a
SS3:
1, s.t.
a
1 =
a
.
SS4:
a
−1
, s.t.
a
a
−1
= 1 for
a
≠ 0
b)
2. a)
A subset W of a vector space V is a subspace of V if W is a vector space, where addition and
scalar multiplication of vectors in W produce the same vectors as these operations did in V.
b)
Nonempty: Since
W
1
and
W
2
are subspaces of
V,
0
W
1
,
0
W
2
, then
W
W
W
2
1
0
Closure under addition:
2
1
,
W
W
W
v
u
,
1
,
W
v
u
and
2
,
W
v
u
.
Since
W
1
and
W
2
are subspaces of
V,
1
W
v
u
and
2
W
v
u
.
Therefore,
2
1
W
W
W
v
u
.
Closure under scalarmultiplication:
W
u
,and
R
r
1
W
u
and
2
W
u
.
Since
W
1
and
W
2
are subspaces of
V,
1
W
u
r
and
2
W
u
r
.
Therefore,
2
1
W
W
W
u
r
.
So
W
is a subspace of V.
3. 1)
Since
T(1) = 0,
T(x) = x,
T(x
2
) = 2 + 2x
2
= -4(x) + 2(1 + x) + 2(x + x
2
),
T(x
3
) = 6x + 3x
3
,
The matrix representation of T with respect to B is
3
0
0
0
0
2
0
0
0
2
0
0
6
4
1
0
A
.
2)
B
x
x
x
T
3
2
2
3
4
3
4
4
1
1
2
3
4
3
0
0
0
0
2
0
0
0
2
0
0
6
4
1
0
Therefore, T(4 + 3x + 2x
2
+ x
3
) = 1(x) + 4(1+ x) + 4(x + x
2
) + 3x
3
= 4 + 9x + 4x
2
+ 3x
3
0
1
t
1+ t
0
0
0
0
0
1
0
1
t
1+ t
t
0
t
1+ t
1
1+ t
0
1+ t
1
t

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2
4.
1) F
2) T
3) F
4) F
5) T
6) T
7) T
5.
1) b
2) c
3) d
4) d
5) a
6.
Yes.
Let
3
3
2
2
1
1
1
,
1
,
1
t
t
t
t
t
t
V and
R
s
r
,
.
V
t
t
t
t
t
t
t
t
2
1
2
1
2
2
1
1
1
1
1
and
V
rt
rt
t
t
r
1
1
1
1
1
1
A1:

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