140 CHAPTER 8. MATRICES

cos,sin the coordinates of T (e1) are as shown in the picture. The coordinates of T (e2) alsofollow from simple trigonometry. Thus

Te1 =

(cosθ

sinθ

),Te2 =

(−sinθ

cosθ

).

Therefore, from Theorem 8.3.2,

A =

(cosθ −sinθ

sinθ cosθ

)

For those who prefer a more algebraic approach, the definition of (cos(θ) ,sin(θ)) isas the x and y coordinates of the point (1,0) . Now the point of the vector from (0,0) to(0,1), e2 is exactly π/2 further along along the unit circle. Therefore, when it is rotatedthrough an angle of θ the x and y coordinates are given by

(x,y) = (cos(θ +π/2) ,sin(θ +π/2)) = (−sinθ ,cosθ) .

Example 8.5.2 Find the matrix of the linear transformation which is obtained by first ro-tating all vectors through an angle of φ and then through an angle θ . Thus you want thelinear transformation which rotates all angles through an angle of θ +φ .

Let Tθ+φ denote the linear transformation which rotates every vector through an angleof θ + φ . Then to get Tθ+φ , you could first do Tφ and then do Tθ where Tφ is the lineartransformation which rotates through an angle of φ and Tθ is the linear transformationwhich rotates through an angle of θ . Denoting the corresponding matrices by Aθ+φ , Aφ ,and Aθ , you must have for every x

Aθ+φx= Tθ+φx= Tθ Tφx= Aθ Aφx.

Consequently, you must have

Aθ+φ =

(cos(θ +φ) −sin(θ +φ)

sin(θ +φ) cos(θ +φ)

)= Aθ Aφ

=

(cosθ −sinθ

sinθ cosθ

)(cosφ −sinφ

sinφ cosφ

).

You know how to multiply matrices. Do so to the pair on the right. This yields(cos(θ +φ) −sin(θ +φ)

sin(θ +φ) cos(θ +φ)

)

=

(cosθ cosφ − sinθ sinφ −cosθ sinφ − sinθ cosφ

sinθ cosφ + cosθ sinφ cosθ cosφ − sinθ sinφ

).

Don’t these look familiar? They are the usual trig. identities for the sum of two anglesderived here using linear algebra concepts.

You do not have to stop with two dimensions. You can consider rotations and othergeometric concepts in any number of dimensions. This is one of the major advantages