Special Linear Transformations in ℝ²
Special examples of linear transformations in ℝ² include rotations and reflections.
Geometric descriptions of vector addition and scalar multiplication are used to show that a rotation of vectors through an angle and that a reflection of a vector across a line are examples of linear transformations.
Theorem of Rotation Consider the following diagram:
To obtain T(u + v), T(u) and T(v) are added. This is because, if you add T(u) to T(v), the result is the diagonal of the parallelogram determined by T(u) and T(v). If vectors u and v were added first and then apply the transformation T to the u + v vector, the result is T(u + v), which is the same vector as T(u) + T(v). Therefore, T(u + v) = T(u) + T(v).
This is because the rotation by T preserves all angles between the vectors. It preserves the shape of the parallelogram determined by the two vectors. T distributes across addition of the vectors in ℝ².
If k is a scalar, then T(ku) = kT(u). Then not only does the rotation by T preserve all angles between the vector, it also preserves their magnitude.
Therefore, rotations are an example of a linear transformation, where a matrix of a linear transformation can be found that rotates all vectors through an angle of θ.
Let Rθ : ℝ² -> ℝ² be a linear transformation given by rotating vectors through an angle of θ. Then the matrix A of Rθ is the following:
The following is the function of Rθ for x ∈ ℝ²:
Rθ(x) = Ax
where A is the matrix defined above.
Let e₁ = [1 0]T and e₂ = [0 1]T. These are the geometric vectors which lie on the positive x-axis and the positive y-axis.
Let Rπ/2 : ℝ² -> ℝ² denote the rotation through π/2.
To find the matrix of Rπ/2, use the theorem of the matrix that induces Rπ/2.
To find Rπ/2(x) where x = [1 -2]T, use the function of Rθ.
Theorem of Reflection Let Qm : ℝ² -> ℝ² be a linear transformation given by reflecting vectors over the line y = mx. Then matrix of Qm is the following:
The following is the function of Qm for x ∈ ℝ²:
Qm(x) = Ax
where A is the matrix defined above.
Let Q₂ : ℝ² -> ℝ² denote the reflection over the line y = 2x. Then Q₂ is a linear transformation.
To find the matrix of Q₂, use the theorem of the matrix that induces Q₂.
To find Q₂(x) where x = [1 -2]T, use the function of Qm.
Rotation and Reflection When finding the matrix of the composite transformation Qm ○ Rθ, where matrix A is the matrix of Rθ and matrix B is the matrix of Qm, the matrix of the composite transformation Qm ○ Rθ is BA.
To find the matrix of the composite transformation obtained by first rotating all vectors through an angle of π/6 and then reflecting through the x-axis, start by finding the matrix A of Rπ/6.
Next, find the matrix B of Qm. Since the vectors are reflected through the x-axis, m = 0.
By definition, the matrix of the composite transformation Q₀ ○ Rπ/6 is BA.
