Homework 4  Solutions
2.2 Problem 1
1. TRUE. See Theorem 2.7 (a).
2. TRUE. See last paragraph on page 80.
3. FALSE. It will be a
n
×
m
matrix.
4. TRUE. See Theorem 2.8 (a).
5. TRUE. See Theorem 2.7.
6. FALSE.
L
(
V, W
) is a collection of functions from
V
to
W
, and
L
(
W, V
) is a collection of functions
from
W
to
V
. These are not the same unless
V
and
W
are the same.
2.2 Problem 4.
T
1
0
0
0
= 1
T
0
1
0
0
= 1 +
x
2
T
0
0
1
0
= 0
T
0
0
0
1
= 2
x
The coordinates of these four polynomials with respect to the basis
β
form the four columns of the matrix
[
T
]
γ
β
:
[
T
]
γ
β
=
⎛
⎝
1
1
0
0
0
0
0
2
0
1
0
0
⎞
⎠
2.2 Problem 9.
Let
z
1
=
a
1
+
ib
1
,
z
2
=
a
2
+
ib
2
, and
α
∈
R
. . Then
T
(
z
1
+
αz
2
) =
T
((
a
1
+
ib
1
) +
α
(
a
2
+
ib
2
))
=
T
((
a
1
+
αa
2
) +
i
(
b
1
+
αb
2
))
= (
a
1
+
αa
2
)
−
i
(
b
1
+
αb
2
)
= (
a
1
−
ib
1
) +
α
(
a
2
−
ib
2
)
=
z
1
+
α
z
2
=
T
(
z
1
) +
αT
(
z
2
)
This proves T is linear.
T
(1) = 1, and
T
(
i
) =
−
i
. In the basis
{
1
, i
}
, the coordinates for
T
(1) and
T
(
i
) are (1
,
0) and (0
,
−
1)
(written as columns). Therefore,
[
T
]
β
=
1
0
0
−
1
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2.2 Problem 10.
The
j
th
column of [
T
]
β
is obtained by expressing
T
(
v
j
) in the basis
β
. Since
T
(
v
j
) =
v
j
+
v
j

1
, we get
T
(
v
j
) = 0
v
1
+
· · ·
+ 1
v
j

1
+ 1
v
j
· · ·
0
v
n
The coordinates for
T
(
v
j
) in the basis
β
are (0
, . . . ,
1
,
1
, . . .,
0), where the only nonzero entries are at the
j
−
1 and
j
positions (written as a column vector). For the special case
j
= 1, the only nonzero entry is the
first entry (since
v
j

1
is taken to be 0). Therefore,
[
T
]
β
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1
1
0
0
· · ·
0
0
1
1
0
· · ·
0
0
0
1
1
· · ·
0
.
.
.
0
0
· · ·
0
1
1
0
· · ·
0
1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
This is a matrix which has (a) 1 on each diaginal entry and each entry above a diagonal entry (b) zero in
all other entries.
2.2 Problem 16.
Let the dimension of
V
and
W
be
n
. Assume that the rank of
T
is
k
. Let
S
=
{
v
1
, . . . , v
n

k
}
be a basis
of
N
(
T
). Extend
S
to a basis
β
=
{
v
1
, . . . , v
n

k
, v
n

k
+1
, . . . , v
n
}
of
V
(by Corollary on Page 51). Define
w
j
=
T
(
v
j
) for
j
=
n
−
k
+ 1
, . . . , n
. Let
R
=
{
w
n

k
+1
, . . . , w
n
}
.
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 Winter '07
 Liu
 Linear Algebra, Algebra, Rank, basis, VJ

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