Partial Solution Set, Leon
§
4.3
4.3.2 Let [
u
1
,
u
2
] and [
v
1
,
v
2
] be ordered bases for
R
2
, where
u
1
= (1
,
1)
T
,
u
2
= (

1
,
1)
T
,
v
1
= (2
,
1)
T
, and
v
2
= (1
,
0)
T
. Let
L
be the linear transformation deﬁned by
L
(
x
) =
(

x
1
,x
2
)
T
, and let
B
be the matrix representing
L
with respect to [
u
1
,
u
2
].
{
Note:
B
was actually part of problem 1 in this chapter. As usual, the ﬁrst column of
B
is
[
L
(
u
1
)]
U
= (0
,
1)
T
, and the second column of
B
is [
L
(
u
)
2
]
U
= (1
,
0)
T
.
}
(a) Find the transition matrix
S
corresponding to the change of basis from [
u
1
,
u
2
] to
[
v
1
,
v
2
].
Solution
: The transition matrix in question is the one I’ve been calling
T
UV
, i.e.,
S
=
V

1
U
=
±
0
1
1

2
²±
1

1
1
1
²
=
±
1
1

1

3
²
.
(b) Find the matrix
A
representing
L
with respect to [
v
1
,
v
2
] by computing
A
=
SBS

1
.
Solution
: First we ﬁnd
S

1
=
1
2
±
3
1

1

1
²
. Then it is a simple matter to
determine that
A
=
SBS

1
=
±
1
0

4

1
²
.
4.3.3 Let
L
be the linear transformation on
R
3
given by
L
(
x
) = (2
x
1

x
2

x
3
,
2
x
2

x
1

x
3
,
2
x
3

x
1

x
2
)
T
,
and let
A
be the matrix representing
L
with respect to the standard basis for
R
3
. If
u
1
= (1
,
1
,
0)
T
,
u
2
= (1
,
0
,
1)
T
, and
u
3
= (0
,
1
,
1)
T
, then [
u
1
,
u
2
,
u
3
] is an ordered basis
for
R
3
.