Solutions to Homework 5.
Math 110, Fall 2006.
Prob 2.5.3.
(
a
)
a
2
b
2
c
2
a
1
b
1
c
1
a
0
b
0
c
0
,
(
b
)
a
0
b
0
c
0
a
1
b
1
c
1
a
2
b
2
c
2
,
(
c
)
0

1
0
1
0
0

3
2
1
,
(
d
)
2
1
1
3

2
1

1
3
1
,
(
e
)
5

6
3
0
4

1
3

1
2
,
(
f
)

2
1
2
3
4
1

1
5
2
.
Prob 2.5.6.
The matrix
Q
is simply the change of basis matrix where the change is from the standard
basis to the basis
β
. That is,
Q
is the matrix whose columns are vectors in
β
.
(a)
[
L
A
]
β
=
±
6
11

2

4
²
,
Q
=
±
1
1
1
2
²
.
(b)
[
L
A
]
β
=
±
3
0
0

1
²
,
Q
=
±
1
1
1

1
²
.
(c)
[
L
A
]
β
=
2
2
2

2

3

4
1
1
2
,
Q
=
1
1
1
1
0
1
1
1
2
.
(d)
[
L
A
]
β
=
6
0
0
0
12
0
0
0
18
,
Q
=
1
1
1
1

1
1

2
0
1
.
Prob 2.5.7.
(a)
By the linearity of
T
,
T
(
x,y
) = (
ax
+
by,cx
+
dy
) for some
a
,
b
,
c
,
d
. To ﬁnd these numbers, notice
that every vector on the line remains unchanged under the action of
T
, i.e., (
ax
+
bmx,cx
+
dmx
) =
T
(
x,mx
) = (
x,mx
) for all
x
. This implies
a
+
bm
= 1,
c
+
dm
=
m
. Now, every vector perpendicular
to the line gets reﬂected about the line, i.e., (
amx

bx,cmx

dx
) =
T
(