518 Chapter 18Matrices and linear transformations
(c)Cchanges the sign of the ycoordinate,
and represents reflection in the x-axis.
(d)Dmultiplies the ycoordinate by factor a,
and represents a stretch in the ydirection ifa 1 > 11 and a contraction if 01 < 1 a 1 < 11.
0 Exercises 54, 55
Simultaneous transformations
A coordinate transformation can be applied simultaneously to more than one point if
the column matrix of the coefficients of one point is replaced by the rectangular
matrix whose columns are the coordinates of several points. Thus, ifx
1, x
2, x
3, =, x
nare column vectors for npoints in a three-dimensional space, we construct the matrix
(18.49)
IfAx
i1 = 1 x′
ithen
AX 1 = 1 A(x
1x
2x
3- x
n)
= 1 (Ax
1Ax
2Ax
3- Ax
n)
= 1 (x′
1x′
2x′
3- x′
n)
= 1 X′ (18.50)
and the columns ofX′are the coordinates of the transformed points.
Xxxx x==
()
1231
23123123nnnnxxx x
yyy y
zzz z
10
0 a
x
y
x
ay
=
10
01 −
=
−
x
y
x
y
..........................θ
(x,y)
(x
′
,y
′
)
o
y
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A
(x,y)
(y,x)
x=y
o
y
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(x,y)
(x,−y)
o
y
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..................................C
(x,y)
(x,ay)
o
x
y
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..D
Figure 18.3