A First Course in Linear Algebra, 2017a

282 Linear Transformations
More generally, denote a transformation given by a rotation by T . Why is such a transformation linear?
Consider the following picture which illustrates a rotation. Let ⃗u,⃗v denote vectors.
T (⃗u)+T(⃗v)
T (⃗v)
T (⃗v)
T (⃗u)
⃗v
⃗v
⃗u +⃗v
⃗u
Let’s consider how to obtain T (⃗u +⃗v). Simply, you add T (⃗u) and T (⃗v). Here is why. If you add
T (⃗u) to T (⃗v) you get the diagonal of the parallelogram determined by T (⃗u) and T (⃗v), as this action is our
usual vector addition. Now, suppose we first add ⃗u and ⃗v, and then apply the transformation T to ⃗u +⃗v.
Hence, we find T (⃗u +⃗v). As shown in the diagram, this will result in the same vector. In other words,
T (⃗u +⃗v)=T (⃗u)+T(⃗v).
This is because the rotation preserves all angles between the vectors as well as their lengths. In particular,
it preserves the shape of this parallelogram. Thus both T (⃗u)+T (⃗v) and T (⃗u +⃗v) give the same
vector. It follows that T distributes across addition of the vectors of R 2 .
Similarly, if k is a scalar, it follows that T (k⃗u) =kT (⃗u). Thus rotations are an example of a linear
transformation by Definition 9.55.
The following theorem gives the matrix of a linear transformation which rotates all vectors through an
angle of θ.
Theorem 5.22: Rotation
Let R θ : R 2 → R 2 be a linear transformation given by rotating vectors through an angle of θ. Then
the matrix A of R θ is given by [ cos(θ)
] −sin(θ)
sin(θ) cos(θ)
[ ] [
1 0
Proof. Let⃗e 1 = and⃗e
0 2 =
1
x axis and positive y axis as shown.
]
. These identify the geometric vectors which point along the positive
(−sin(θ),cos(θ))
R θ (⃗e 2 )
θ
⃗e 2
R θ (⃗e 1 )
θ
(cos(θ),sin(θ))
⃗e 1