Midterm Solutions
J. Scholtz
October 29, 2006
Question 1: True vs. False
1.
Question:
Let
W
1
,
W
2
be subspaces of
V
and
β
1
,
β
2
be their respective basis. Then is it true that
β
1
∩
β
2
is a basis for
W
1
∩
W
2
?
Answer:
No, consider
W
1
=
W
2
=
R
⊂
R
. Let
β
1
=

1 and
β
2
= 1. Then
β
1
∩
β
2
= 0 and so it
is not a basis for
W
1
∩
W
2
=
R
.
2.
Question:
Let
P
(
R
) denote the infinite dimensional vector space of all polynomials with real
ceficients. Let
g
(
x
)
∈
P
(
R
) be some fixed polynomial. Then the function
T
:
P
(
R
)
→
P
(
R
), such
that
T
(
f
(
x
)) =
g
(
x
)
f
(
x
) is linear.
Answer:
Yes, we can check that
T
(
c
×
a
(
x
) +
b
(
x
)) =
g
(
x
)(
c
×
a
(
x
) +
b
(
x
)) =
g
(
x
)(
c
×
a
(
x
)) +
g
(
x
)
b
(
x
) =
c
×
T
(
a
(
x
)) +
T
(
b
(
x
)).
3.
Question:
Let
V
be a vector space and
T, U
:
V
→
V
be two linear operators. Then
N
(
T
)
⊂
N
(
TU
).
Answer:
No.
Consider
V
=
R
2
and the maps
T
(
x, y
) = (
x,
0) and
U
(
x, y
) = (
y, x
).
Then
N
(
T
) = span((0
,
1)), whereas
N
(
TU
) = span((1
,
0)).
4.
Question:
Let
V
be a vector space and
T, U
:
V
→
V
be two linear operators. Then
R
(
TU
)
⊂
R
(
T
).
Answer:
Yes.
R
(
TU
) =
T
(
U
(
V
))) =
T
(
R
(
U
))
⊂
T
(
V
) since
U
(
V
) =
R
(
U
)
⊂
V
.
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 Summer '08
 GUREVITCH
 Linear Algebra, Algebra, J. Scholtz

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