Page 188 - A Course in Linear Algebra with Applications

172 Chapter Six: Linear Transformationns

Example 6.2.10
Consider the change of basis in R 2 which arises when the x-
and y-axes are rotated through angle 9 in an anticlockwise
direction. As was noted in Example 6.2.6, the effect of this
rotation is to replace the standard ordered basis B = {E\, E2]
by the basis B' consisting of

cos 9 \ _. / — sin 9
sin 9 J y cos 9

The matrix which describes the change of basis B ' —> B is

q _ f cos 9 — sin 9
\ sin 9 cos 9

so the change of basis B —> B' is described by

_
,_i
s- l = / cos 9 sin 9
— sin 9 cos 9

Hence, if X = I , I, the coordinate of vector of X with re-

spect to the basis B' is

c
r v\ — -i v _ f a cos 9 + b sin 9 \

This means that the coordinates of the point (a, b) with re-
spect to the rotated axes are

a' = a cos 9 + b sin # and b' = — a sin 9 + b cos 9,

respectively.