A First Course in Linear Algebra, 2017a

5.4. Special Linear Transformations in R 2 283
From Theorem 5.6, we need to find R θ (⃗e 1 ) and R θ (⃗e 2 ), and use these as the columns of the matrix A
of T . We can use cos,sin of the angle θ to find the coordinates of R θ (⃗e 1 ) as shown in the above picture.
The coordinates of R θ (⃗e 2 ) also follow from trigonometry. Thus
R θ (⃗e 1 )=
[ cosθ
sinθ
]
,R θ (⃗e 2 )=
[ −sinθ
cosθ
]
Therefore, from Theorem 5.6,
[ ]
cosθ −sinθ
A =
sinθ cosθ
We can also prove this algebraically without the use of the above picture. The definition of (cos(θ),sin(θ))
is as the coordinates of the point of R θ (⃗e 1 ). Now the point of the vector⃗e 2 is exactly π/2 further along the
unit circle from the point of ⃗e 1 , and therefore after rotation through an angle of θ the coordinates x and y
of the point of R θ (⃗e 2 ) are given by
Consider the following example.
Example 5.23: Rotation in R 2
(x,y)=(cos(θ + π/2),sin(θ + π/2)) = (−sinθ,cosθ)
Let R π
2
: R 2 → R 2 denote rotation through π/2. Find the matrix of R π
2
. Then, find R π
2
(⃗x) where
[ ]
1
⃗x = .
−2
♠
Solution. By Theorem 5.22, the matrix of R π is given by
2
[ ] [ cos(θ) −sin(θ) cos(π/2) −sin(π/2)
=
sin(θ) cos(θ) sin(π/2) cos(π/2)
]
=
[ 0 −1
1 0
]
To find R π
2
(⃗x), we multiply the matrix of R π
2
by⃗x as follows
[ 0 −1
1 0
][
1
−2
] [ 2
=
1
]
♠
We now look at an example of a linear transformation involving two angles.
Example 5.24: The Rotation Matrix of the Sum of Two Angles
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. Hence the linear transformation rotates all vectors through
an angle of θ + φ.