(a) Prove that the linear transformation associated with the...

(a) Prove that the linear transformation associated with the improper orthogonal matrix $\left(\begin{array}{rr}\cos \theta & \sin \theta \\ \sin \theta & -\cos \theta\end{array}\right)$ is a reflection through the line that makes an angle $\frac{1}{2} \theta$ with the $x$-axis. (b) Show that the composition of two such reflections, with angles $\theta, \varphi$, is a rotation. What is the angle of the rotation? Does the composition depend upon the order of the two reflections?

(a) Prove that the linear transformation associated with the improper orthogonal matrix $\left(\begin{array}{rr}\cos \theta & \sin \theta \\ \sin \theta & -\cos \theta\end{array}\right)$ is a reflection through the line that makes an angle $\frac{1}{2} \theta$ with the $x$-axis.
(b) Show that the composition of two such reflections, with angles $\theta, \varphi$, is a rotation.
What is the angle of the rotation? Does the composition depend upon the order of the two reflections?

Key Concepts

Order of Reflections

The composition of two reflections leading to a rotation has the property that the resulting rotation is independent of the order in which the reflections are applied. This is because rotation by a given angle is commutative; thus, reflecting first in one line and then in another will result in the same rotation as if the order were reversed. This order-independence is an important feature in the study of plane isometries.

Reflection in the Plane

A reflection in the plane is a linear transformation that “flips” each point to the other side of a given line (the axis of reflection) while keeping points on the line fixed. Such transformations are represented by matrices that are orthogonal but not proper, since their determinant is -1. Understanding reflections is key to analyzing how geometric figures are altered by such linear maps.

Improper Orthogonal Matrices

Improper orthogonal matrices are those orthogonal matrices whose determinant is -1. They include reflections and other isometries that reverse orientation. Recognizing that a given matrix is improper helps in understanding the transformation’s action and its geometric interpretation, particularly when distinguishing between rotations (which preserve orientation) and reflections (which do not).

Rotation Transformation

A rotation transformation in the plane is an isometry that rotates every point about a fixed point (commonly the origin) by a given angle, preserving the distance between points. Rotations are represented by proper orthogonal matrices (determinant 1) and are a central concept in understanding the composition of other isometries such as reflections.

Composition of Reflections

A key geometric fact is that the composition of two reflections in the plane is always a rotation. More precisely, if two reflections are taken across lines that intersect at the origin, the composition yields a rotation by twice the angle between the two lines. This insight connects seemingly different isometries and is a powerful tool for understanding their combined effects.

Angle Calculation in Composite Transformations

When two reflections are composed, the rotation angle produced is twice the difference between the angles that the reflection axes make with a fixed line (such as the x-axis). This doubling arises from the fact that each reflection changes the angle by the same amount, and so their composition magnifies the effect. Such calculations are essential for determining the exact nature of the transformation.

Please give Ace some feedback