SSM:
Elementary Linear Algebra
Section 4.11
113
(c)
A reflection through the
yz
-plane maps
(1, 0, 0) to (
−
1, 0, 0), (0, 1, 0) to (0, 1, 0),
and (0, 0, 1) to (0, 0, 1), so the standard
matrix is
100
010
001
.
−
⎡⎤
⎢⎥
⎣⎦
5. (a)
Rotation of 90
°
about the
z
-axis maps
(1, 0, 0) to (0, 1, 0), (0, 1, 0) to (
−
1, 0, 0),
and (0, 0, 1) to (0, 0, 1), so the standard
matrix is
01
0
.
−
(b)
Rotation of 90
°
about the
x
-axis maps
(1, 0, 0) to (1, 0, 0), (0, 1, 0) to (0, 0, 1), and
(0, 0, 1) to (0,
−
1, 0), so the standard matrix
is
10 0
00 1
01 0
.
−
(c)
Rotation of 90
°
about the
y
-axis maps
(1, 0, 0) to (0, 0,
−
1), (0, 1, 0) to (0, 1, 0),
and (0, 0, 1) to (1, 0, 0), so the standard
matrix is
.
−
7.
300
0
0
,
−
⎡
⎤
⎡⎤ ⎡⎤
=
⎢
⎥
⎢⎥ ⎢⎥
⎣
⎦
⎣⎦ ⎣⎦
301
3
0
,
−−
⎡
⎤⎡ ⎤
⎡
⎤
=
⎢
⎥⎢ ⎥
⎢
⎥
⎣
⎦⎣ ⎦
⎣
⎦
0
011
1
,
−
⎡
⎤
=
⎢
⎥
⎣
⎦
3
1
⎡
⎤
⎡
⎤⎡⎤
=
⎢
⎥
⎢
⎥⎢⎥
⎣
⎦
⎣
⎦⎣⎦
The image is a rectangle with vertices at (0, 0),
(
−
3, 0), (0, 1), and (
−
3, 1).
y
x
9. (a)
Shear by a factor of
k
= 4 in the
y
-direction
has matrix
10
41
.
(b)
Shear by a factor of
k
=
−
2 in the
x
-direction
has matrix
12
.
−
11. (a)
The geometric effect is expansion by a
factor of 3 in the
x
-direction.
(b)
The geometric effect is expansion by a
factor of 5 in the
y
-direction and reflection
about the
x
-axis.
(c)
The geometric effect is shearing by a factor
of 4 in the
x
-direction.
13. (a)
11
22
00
05
⎡
⎤⎡
⎤
=
⎢
⎥⎢
⎥
⎣
⎦⎣
⎦
(b)
10 10
2105
25
⎡
⎤
⎡
⎤
=
⎢
⎥
⎢
⎥
⎣
⎦
⎣
⎦
(c)
180
180
0
1
180
180
1 0
1001
0
cos
sin
sin
cos
°−
°
⎡
⎤
⎢
⎥
°°
⎣
⎦
−
⎡
⎤
=
⎢
⎥
−
⎣
⎦
−
=
−
15. (a)
Since
1
,
−
⎡
=
⎢
⎣
the inverse
transformation is reflection about
y
=
x
.
(b)
Since for 0 <
k
< 1,
1
1
0
0
k
k
−
⎡
⎤
=
⎢
⎥
⎣
⎦
and
1
1
0
0
,
k
k
−
⎡
⎤
=
⎢
⎥
⎣
⎦
the inverse
transformation is an expansion along the
same axis.
(c)
Since
1
−
⎡
⎤
=
⎢
⎥
⎣
⎦
and
1
−
⎡
⎤
=
⎢
⎥
⎣
⎦
the inverse
transformation is a reflection about the same
coordinate axis.
(d)
Since
k
≠
0,
1
0 1
kk
−
−
⎡⎤⎡
⎤
=
⎢⎥⎢
⎥
⎣⎦⎣
⎦
and
1
,
−
⎡
⎤
=
⎢
⎥
−
⎣
⎦
the inverse
transformation is a shear (in the opposite
direction) along the same axis.