Page 181 - A Course in Linear Algebra with Applications

6.2: Linear Transformations and Matrices 165

Consequently

T{ \x 2 ) = A \x 2
\x 3J \x 3

as can be verified directly by matrix multiplication.
Example 6.2.7
Consider the linear operator T : R 2 —> R 2 which arises from
an anti-clockwise rotation in the xy-plane through an angle 9
(see Example 6.2.2.) The problem is to write down the matrix
which represents T.
All that need be done is to identify the vectors T(E{)
and T(E 2) where E\ and E 2 are the vectors of the standard
ordered basis.

-(0,1)
(-sin 9, cos 9)

(cos 9, sin 9)

(1.0)

The line segment representing E\ is drawn from the origin O to
the point (1, 0), and after rotation it becomes the line segment

from O to the point (cos 8, sin 9); thus T(E\) = [ . n 1.
sin
\ " J
— sin 9
Similarly T(E 2) = It follows that the matrix
cos 9
which represents the rotation T is
cos 9 -sin 9
sin 9 cos 9