18.5 Linear transformations 517
(18.47)
that represents the transformation of the vector with components(x
1, x
2, =, x
n)into
the new vector(x′
1, x′
2, =, x′
n).In matrix notation,
x′ 1 = 1 Ax (18.48)
where
The n-dimensional vectors are represented by the column matrices xand x′, and Ais
called the transformation matrixthat transforms xinto x′.
EXAMPLE 18.18Linear transformations in two dimensions
Let(x, y)be the cartesian coordinates of a point in the xy-plane, and let the matrices
represent transformations in the plane. Then, as illustrated in Figure 18.3,
(a)Ais the rotation through angle θabout the origin discussed in Example 18.1.
(b)Binterchanges the xand ycoordinates,
and represents reflection in (or rotation through 180° about) the linex 1 = 1 y.
01
10
=
x
y
y
x
AB=
−
,=
,
cos sin
sin cos
θθ
θθ
01
10
CCD=
−
,=
10
01
10
0 a
x′=
′
′
′
′
x
x
x
x
n123,=A
aaa a
aaa a
aaa
nn11
12 13121 22 23 231 32 33aa
aaa a
nn
nnnn31
23
,=x
x
x
x
x
n123′=
′=
′=
′=
x
x
x
x
ax
ax
ax
ax
a
nn12311 121 131 1111 12 222 232 22213 323 33x
ax
ax
ax
ax
ax
a
n++
++
3333312x
ax
ax
ax
ax
ax
nnnnnnnnn n++
++
3