SOLUTION TO ASSIGNMENT 6  MATH 251, WINTER 2007
1. Let
T
:
V
→
W
be a linear map and define
T
*
:
W
*
→
V
*
by (
T
*
(
g
))(
v
) :=
g
(
Tv
). Prove the following
lemma:
L
EMMA
1.
(1)
T
*
is a welldefined linear map.
(2)
Let
B, C
be bases to
V, W
, respectively. Let
A
=
C
[
T
]
B
be the
m
×
n
matrix representing
T
, where
n
= dim(
V
)
, m
= dim(
W
)
. Then the matrix representing
T
*
with respect to the dual bases
B
*
, C
*
is the transpose of
A
:
B
*
[
T
*
]
C
*
=
C
[
T
]
B
t
.
(3)
If
T
is injective then
T
*
is surjective. (Do NOT use Proposition 7.2.8 in the notes).
(4)
If
T
is surjective then
T
*
is injective.
Proof.
(1) First, we need to show that the map
v
→
g
(
T
(
v
)) is a linear map. This is easy:
T
and
g
are linear maps and so is their composition. Next, we need to check that
T
*
is linear. Namely,
that for every
v
,
a
i
∈
F
, g
i
∈
W
*
, we have
(
T
*
(
a
1
g
1
+
a
2
g
2
))(
v
) = (
a
1
T
*
g
1
+
a
2
T
*
g
2
)(
v
)
.
This follows from definitions: (
T
*
(
a
1
g
1
+
a
2
g
2
))(
v
) = (
a
1
g
1
+
a
2
g
2
)(
Tv
) =
a
1
·
g
1
(
Tv
)+
a
2
·
g
2
(
Tv
) =
a
1
·
T
*
g
1
(
v
) +
a
2
·
T
*
g
2
(
v
) = (
a
1
T
*
g
1
+
a
2
T
*
g
2
)(
v
).
(2) Let us write
B
=
{
b
1
, . . . , b
n
}
, C
=
{
c
1
, . . . , c
m
}
, B
*
=
{
b
*
1
, . . . , b
*
n
}
, C
*
=
{
c
*
1
, . . . , c
*
m
}
, where we
have
b
*
i
(
b
j
) =
δ
ij
,
c
*
i
(
c
j
) =
δ
ij
.
If
C
[
T
]
B
=
A
= (
a
ij
) that means that
[
Tb
i
]
C
=
a
1
i
c
1
+
· · ·
+
a
mi
c
m
.
Let us write
T
*
c
*
i
=
α
1
i
b
*
1
+
· · ·
+
α
ni
b
*
n
,
so that
B
*
[
T
*
]
C
*
= (
α
ij
)
,
and we wish to prove that
α
ji
=
a
ij
.
As we have seen before, the coefficient
α
ji
is equal to (
T
*
c
*
i
)(
b
j
) (cf. the solution to question (2)
below; we make use of
B
being the dual basis to
B
*
. See also the notes, Example 7.1.4). But, by
definition, (
T
*
c
*
i
)(
b
j
) =
c
*
i
(
Tb
j
) =
c
*
i
(
a
1
j
c
1
+
· · ·
+
a
mj
c
m
) =
a
ij
.
(3) Suppose
T
:
V
→
W
is injective.
This means that the column rank of
A
=
C
[
T
]
B
is maximal,
equal to
n
, which is the row rank of
A
t
. But then also the column rank of
A
t
is
n
. Therefore, the
image of
T
*
is
n
dimensional. Since dim(
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 Spring '08
 Goren
 Math, Linear Algebra, Linear map, Latin Square, column rank, Latin squares

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