Homework 6 Solutions
Joshua Hernandez
November 11, 2009
2.5 - The Change of Coordinate Matrix
2.
For each of the following pairs of ordered bases
β
and
β
0
for
R
2
, ﬁnd the change of coordinate matrix
that changes
β
0
-coordinates into
β
coordinates.
b.
β
=
{
(-1
,
3)
,
(2
,
-1)
}
and
β
0
=
{
(0
,
10)
,
(5
,
0)
}
.
Solution:
We want to ﬁnd
Q
= [
I
R
2
]
β
β
0
. The usual procedure:
I
R
2
(0
,
10) = (0
,
10) =
a
(-1
,
3) +
b
(2
,
-1)
,
I
R
2
(5
,
0) = (5
,
0) =
c
(-1
,
3) +
d
(2
,
-1)
.
We can write these two systems as matrix equations
±
-1
2
3
-1
²±
a
b
²
=
±
0
10
²
,
±
-1
2
3
-1
²±
c
d
²
=
±
5
0
²
,
or together as
±
-1
2
3
-1
²±
a c
b
d
²
=
±
0
5
10 0
²
.
The matrix (
a c
b d
) will be our change-of-coordinate matrix. We solve by taking inverses:
Q
=
±
a c
b
d
²
=
±
-1
2
3
-1
²
-1
±
0
5
10 0
²
=
1
-5
±
-1 -2
-3 -1
²±
0
5
10 0
²
=
1
-5
±
-20
-5
-10 -15
²
=
±
4 1
2 3
²
.
Note:
In general, let
V
be some ﬁnite-dimensional vector space, let
α
the standard basis of
V
,
and let
β
and
β
0
be two other bases of
V
. Then
[
I
V
]
β
β
0
= [
I
V
]
β
α
[
I
V
]
α
β
0
= ([
I
V
]
α
β
)
-1
[
I
V
]
α
β
0
The matrices [
I
V
]
α
β
0
and [
I
V
]
α
β
are easily computed; one can simply read oﬀ the coeﬃcients from
the basis vectors.
d.
β
=
{
(-4
,
3)
,
(2
,
-1)
}
and
β
0
=
{
(2
,
1)
,
(-4
,
1)
}
.
Solution:
Proceeding as above,
Q
=
±
-4
2
3
-1
²
-1
±
2 -4
1
1
²
=
1
-2
±
-1 -2
-3 -4
²±
2 -4
1
1
²
=
1
-2
±
-4
2
-10 8
²
=
±
2 -1
5 -4
²
.
3.
For each of the following pairs of ordered bases
β
and
β
0
for
P
2
(
R
), ﬁnd the change of coordinate matrix
that changes
β
0
-coordinates into
β
-coordinates.
b.
β
=
{
1
,x,x
2
}
, and
β
0
=
{
a
2
x
2
+
a
1
x
+
a
0
,b
2
x
2
+
b
1
x
+
b
0
,c
2
x
2
+
c
1
x
+
c
0
}
1