Calculus of One and Several Variables

18.3 Rotations in ℝ²

Sometimes you need to find a matrix which represents a given linear transformation which is described in geometrical terms. The idea is to produce a matrix which you can multiply a vector by to get the same thing as some geometrical description. A good example of this is the problem of rotation of vectors discussed above. Consider the problem of rotating through an angle of θ.

Let e₁ ≡

and e₂ ≡. These identify the geometric vectors which point along the positive x axis and positive y axis as shown.

From the above, you only need to find Te₁ and Te₂, the first being the first column of the desired matrix, A and the second being the second column. From the definition of the cos,sin the coordinates of T(e₁) are as shown in the picture. The coordinates of T(e₂) also follow from simple trigonometry. Thus

( ) ( ) Te1 = cosθ ,Te2 = − sinθ . sinθ cosθ

Therefore, from Theorem 18.2.4,

( cosθ − sin θ ) A = sinθ cosθ

For those who prefer a more algebraic approach, the definition of

is as the x and y coordinates of the point . Now the point of the vector from to , e₂ is exactly π∕2 further along along the unit circle. Therefore, when it is rotated through an angle of θ the x and y coordinates are given by

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

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+ ϕx = Tθ+ϕx = TθTϕx = AθA ϕx. (18.10)

(18.10)

Consequently, you must have

You know how to multiply matrices. Do so to the pair on the right. This yields Don’t these look familiar? They are the usual trig. identities for the sum of two angles derived here using linear algebra concepts.

You do not have to stop with two dimensions. You can consider rotations and other geometric concepts in any number of dimensions. This is one of the major advantages of linear algebra. You can break down a difficult geometrical procedure into small steps, each corresponding to multiplication by an appropriate matrix. Then by multiplying the matrices, you can obtain a single matrix which can give you numerical information on the results of applying the given sequence of simple procedures. That which you could never visualize can still be understood to the extent of finding exact numerical answers. Notice how we composed the linear transformations in the above and multiplied the matrices to get the same thing. This is how we do it in general.