Partial Solution Set, Leon
§
3.5
3.5.2
(c) Let
u
1
= (0
,
1)
T
and
u
2
= (1
,
0)
T
. Then
{
u
1
,
u
3
}
is an ordered basis for
R
2
. Find
the transition matrix corresponding to the change of basis from the standard basis to
{
u
1
,
u
3
}
.
Solution
: Let
U
=
{
u
1
u
2
}
. Then
U
is the transition matrix corresponding to the change
of basis from
{
u
1
,
u
3
}
to the standard basis. It follows that
U

1
is the matrix that we’re
after. Performing the computation, we ﬁnd that
U

1
=
1
det
(
T
)
±
0

1

1
0
²
=
U
, i.e.,
U
is its own inverse.
3.5.4
Let
E
=
{
(5
,
3)
T
,
(3
,
2)
T
}
, and let
x
= (1
,
1)
T
,
y
= (1
,

1)
T
, and
z
= (10
,
7)
T
. Find the
values of [
x
]
E
, [
y
]
E
, and [
z
]
E
.
Solution
: From our discussion in class, we know that the transition matrix from the
ordered basis
E
to the standard basis is just
T
=
±
5 3
3 2
²
.
Since we want to go the other way, the transition matrix we want is
W
=
T

1
=
±
2

3

3
5
²
.
It is now an easy step to determine [
x
]
E
= (

1
,
2)
T
, [
y
]
E
= (5
,

8)
T
, and [
z
]
E
= (

1
,
5)
T
.
3.5.5
Let
u
1
= (1
,
1
,
1)
T
,
u
2
= (1
,
2
,
2)
T
, and
u
3
= (2
,
3
,
4)
T
. Let
e
i
denote the
i
th standard
basis vector. In part (a) of this problem, we are to ﬁnd the transition matrix corre
sponding to the change of basis from
{
e
1
,
e
2
,
e
3
}
to