Linear Algebra, Theory And Applications, 2012a

9.3. THE MATRIX OF A LINEAR TRANSFORMATION 235
where T e i is the i th column of A.
Example 9.3.19 Determine the matrix for the transformation mapping R 2 to R 2 which
consists of rotating every vector counter clockwise through an angle of θ.
( )
( )
1 0
Let e 1 ≡ and e
0
2 ≡ . These identify the geometric vectors which point
1
along the positive x axis and positive y axis as shown.
e 2
✻
From Theorem 9.3.18, you only need to find T e 1 and T e 2 , the first being the first column
of the desired matrix A and the second being the second column. From drawing a picture
and doing a little geometry, you see that
T e 1 =
( cos θ
sin θ
✲
e 1
) ( − sin θ
,Te 2 =
cos θ
)
.
Therefore, from Theorem 9.3.18,
( cos θ − sin θ
A =
sin θ cos θ
)
Example 9.3.20 Find the matrix of the linear transformation which is obtained by first
rotating all vectors through an angle of φ and then through an angle θ. Thus you want the
linear transformation which rotates all angles through an angle of θ + φ.
Let T θ+φ denote the linear transformation which rotates every vector through an angle
of θ + φ. Then to get T θ+φ , you could first do T φ and then do T θ where T φ is the linear
transformation which rotates through an angle of φ and T θ is the linear transformation
which rotates through an angle of θ. Denoting the corresponding matrices by A θ+φ , A φ ,
and A θ , you must have for every x
A θ+φ x = T θ+φ x = T θ T φ x = A θ A φ x.
Consequently, you must have
A θ+φ =
=
( )
cos (θ + φ) − sin (θ + φ)
= A
sin (θ + φ) cos (θ + φ)
θ A φ
( )( )
cos θ − sin θ cos φ − sin φ
.
sin θ cos θ sin φ cos φ
Therefore,
(
cos (θ + φ) − sin (θ + φ)
sin (θ + φ) cos (θ + φ)
) (
cos θ cos φ − sin θ sin φ − cos θ sin φ − sin θ cos φ
=
sin θ cos φ +cosθ sin φ cos θ cos φ − sin θ sin φ
)
.
Don’t these look familiar? They are the usual trig. identities for the sum of two angles
derived here using linear algebra concepts.
Example 9.3.21 Find the matrix of the linear transformation which rotates vectors in
R 3 counterclockwise about the positive z axis.