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
a
b
=
0
10
,
-1
2
3
-1
c
d
=
5
0
,
or together as
-1
2
3
-1
a
c
b
d
=
0
5
10
0
.
The matrix (
a c
b d
) will be our change-of-coordinate matrix. We solve by taking inverses:
Q
=
a
c
b
d
=
-1
2
3
-1
-1
0
5
10
0
=
1
-5
-1
-2
-3
-1
0
5
10
0
=
1
-5
-20
-5
-10
-15
=
4
1
2
3
.
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
-1
2
-4
1
1
=
1
-2
-1
-2
-3
-4
2
-4
1
1
=
1
-2
-4
2
-10
8
=
2
-1
5
-4
.
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