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MATH1111/201011/Tutorial V Solution
1
Tutorial V Suggested Solution
1.
Recall: If
f
:
A
!
B
and
g
:
B
!
C
are two functions, then the
composite
f
±
g
of
f
and
g
is a
function from
A
to
C
de±ned as
f
±
g
(
x
) =
f
(
g
(
x
))
.
(a) Let
S
:
U
!
W
and
T
:
V
!
U
be two linear transformations. Show that
S
±
T
:
V
!
W
is a linear transformation.
(b) Suppose
S;T
:
V
!
V
are two linear operators and let
E
be an ordered basis for
V
. If
A
and
B
are the matrix representation of
S
and
T
w.r.t.
E
(i.e. r.t.
E
and
E
) respectively,
show that the matrix representation of
S
±
T
w.r.t
E
is
AB
.
Ans
.
(a) To show the linearity, we check the following:
S
±
T
(
x
+
y
) =
S
±
T
(
x
+
y
)
²
=
S
±
T
(
x
)+
T
(
y
)
²
=
S
±
T
(
x
)
²
+
S
±
T
(
y
)
²
=
S
±
T
(
x
)+
S
±
T
(
y
).
S
±
T
(
±
x
) =
S
±
T
(
±
x
)
²
=
S
±
±T
(
x
)
²
=
±S
±
T
(
x
)
²
=
±S
±
T
(
x
).
(b) Let
x
2
V
. By de±nition,
S
±
T
(
x
) =
S
(
T
(
x
)). Thus,
[
S
±
T
(
x
)]
E
= [
S
(
T
(
x
))]
E
=
A
[
T
(
x
)]
E
=
AB
[
x
]
E
:
Using the uniqueness of matrix representation, we conclude that
AB
is the matrix repre
sentation.
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View Full DocumentMATH1111/201011/Tutorial V Solution
2
2. Let
T
be a linear operator on
R
3
de±ned by
T
((
x y z
)
T
) = (3
x
+
z
±
2
x
+
y
±
x
+ 2
y
+ 4
z
)
T
:
(a) Find its standard matrix representation and show that it is nonsingular matrix.
(b) Let
E
= [
u
1
;
u
2
;
u
3
] where
u
1
= (1 0 1)
T
,
u
2
= (
±
1 2 1)
T
,
u
3
= (2 1 1)
T
. Find
the matrix representation of
T
w.r.t.
E
. Show that this matrix representation is nonsingu
lar.
Ans
.
(a) The standard matrix representation is
0
@
3
0 1
±
2 1 0
±
1 2 4
1
A
, which is nonsingular by checking its
determinant.
(b) From
0
@
3
0 1
±
2 1 0
±
1 2 4
1
A
±
[
u
1
]
St
[
u
2
]
St
[
u
3
]
St
²
=
0
@
3
0 1
±
2 1 0
±
1 2 4
1
A
0
@
1
±
1 2
0
2
1
1
1
1
1
A
=
0
@
4
±
2
7
±
2
4
±
3
3
9
4
1
A
(
St
denotes the standard ordered basis for
R
3
), we see that as
x
= [
x
]
St
in
R
3
,
T
(
u
1
) = [
T
(
u
1
)]
St
= (4
±
2 3)
T
;
T
(
u
2
) = (
±
2 4 9)
T
;
T
(
u
3
) = (7
±
3 4)
T
:
Then
±
[
T
(
u
1
)]
E
[
T
(
u
2
)]
E
[
T
(
u
3
)]
E
²
=
0
@
1
±
1 2
0
2
1
1
1
1
1
A
±
1
0
@
4
±
2
7
±
2
4
±
3
3
9
4
1
A
is the matrix representation w.r.t
E
.
3. Let
L
:
V
!
V
be a linear operator and
A
be its matrix representation w.r.t. an ordered basis
E
. Suppose
F
is another ordered basis for
V
and det
A
6
= 0. Is the matrix representation w.r.t.
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 Spring '10
 Dr,Li

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