18.5 Linear transformations 519
EXAMPLE 18.19Rotate the square whose corners have positions in the xy-plane
(x, y) 1 = 1 (2, 1), (3, 1), (3, 2), (2, 2)
throughπ 24 about the origin as in Figure 18.4.
Because the transformation matrix is
The coordinates of the corners of the square after rotation are then given by
0 Exercises 56–59
Consecutive transformations
Let the transformation x′ 1 = 1 Ax, equation (18.48), be followed by a second
transformation
x′′ 1 = 1 Bx′ (18.51)
that transformsx′intox′′. Substitution of (18.48) into (18.51) then gives
x′′ 1 = 1 BAx 1 = 1 Cx (18.52)
=
1
2
2
2
1
2
0
3
2
4
2
5
2
4
2
1
2
11
11
2332
1122
1
2
1210
3454
−
=
1
2
11
11
−
cosππ 4412 ==sin ,
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y
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34
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.....................................................................................................................................Figure 18.4