MATH 110: LINEAR ALGEBRA
SPRING 2007/08
PROBLEM SET 6 SOLUTIONS
1.
Let
V
and
W
be ﬁnite dimensional vector spaces over
F
. Let
T
:
V
→
W
be a linear transfor
mation.
(a) If
V
0
is a subspace of
V
, show that the set
T
(
V
0
) :=
{T
(
v
)
∈
W

v
∈
V
0
}
is a subspace of
W
.
Solution.
Let
α
1
,α
2
∈
F
and
w
1
,
w
2
∈ T
(
V
0
). Then there exist
v
1
,
v
2
∈
V
0
such that
w
1
=
T
(
v
1
) and
w
2
=
T
(
v
2
)
.
Since
V
0
is a subspace,
α
1
v
1
+
α
2
v
2
∈
V
0
by Theorem
1.8
. So
α
1
w
1
+
α
2
w
2
=
α
1
T
(
v
1
) +
α
2
T
(
v
2
) =
T
(
α
1
v
1
+
α
2
v
2
)
∈ T
(
V
0
)
.
Hence
T
(
V
0
) is a subspace of
W
by Theorem
1.8
.
(b) If
W
0
is a subspace of
W
, show that the set
T

1
(
W
0
) :=
{
v
∈
V
 T
(
v
)
∈
W
0
}
is a subspace of
V
. [
Note
:
T

1
(
W
0
) is just a notation for the set deﬁned by the
rhs
,
T
is
not necessarily invertible.]
Solution.
Let
α
1
,α
2
∈
F
and
v
1
,
v
2
∈ T

1
(
W
0
). Then there exist
w
1
,
w
2
∈
W
0
such
that
T
(
v
1
) =
w
1
and
T
(
v
2
) =
w
2
.
Since
W
0
is a subspace,
α
1
w
1
+
α
2
w
2
∈
W
0
by Theorem
1.8
. So
T
(
α
1
v
1
+
α
2
v
2
) =
α
1
T
(
v
1
) +
α
2
T
(
v
2
) =
α
1
w
1
+
α
2
w
2
∈
W
0
.
So
α
1
v
1
+
α
2
v
2
∈ T

1
(
W
0
). Hence
T

1
(
W
0
) is a subspace of
V
by Theorem
1.8
.
(c) Deduce that
T
(
V
) = im(
T
) and
T

1
(
{
0
W
}
) = ker(
T
). What are
T
(
{
0
V
}
) and
T

1
(
W
)?
What about
T
(ker(
T
)) and
T

1
(im(
T
))?
Solution.
By deﬁnition,
T
(
V
) =
{T
(
v
)
∈
W

v
∈
V
}
= im(
T
). Also
T

1
(
{
0
W
}
) =
{
v
∈
V
 T
(
v
)
∈ {
0
W
}}
=
{
v
∈
V
 T
(
v
) =
0
W
}
= ker(
T
)
.
Clearly,
T
(
{
0
V
}
) =
{
0
W
}
,
T

1
(
W
) =
V
,
T
(ker(
T
)) =
{T
(
v
)
∈
W

v
∈
ker(
T
)
}
=
{
0
W
}
,
T

1
(im(
T
)) =
{
v
∈
V
 T
(
v
)
∈
im(
T
)
}
=
V.
(d) Let
B
W
=
{
w
1
,...,
w
m
}
be a basis for im(
T
). If
v
1
,...,
v
m
∈
V
are such that
T
(
v
i
) =
w
i
for
i
= 1
,...,m
, show that
span
{
v
1
,...,
v
m
} ⊆ T

1
(im(
T
))
but equality need not hold in general.
Date
: May 5, 2008 (Version 1.0).
1