## A summary of the effects of rotations and reflections

This post summarizes the results of previous posts exploring the effects of the following sequences of linear transformations in the x-y plane:

To review, the linear transformation that rotates vectors through an angle $\theta$ can be represented by the matrix $Q_{\theta} = \begin{bmatrix} \cos \theta&-\sin \theta \\ \sin \theta&\cos \theta \end{bmatrix}$

The linear transformation that reflects vectors in the line through the origin with angle $\varphi$ (the $\varphi$-line) can be represented by the matrix $H_{\varphi} = \begin{bmatrix} 2\cos^2 \varphi - 1&2\cos\varphi\sin\varphi \\ 2\cos\varphi\sin\varphi&2\sin^2 \varphi - 1 \end{bmatrix}$ $= \begin{bmatrix} \cos 2\varphi&\sin 2\varphi \\ \sin 2\varphi&-\cos 2\varphi \end{bmatrix}$

Combining these operations (for example, to do a rotation followed by a reflection) is done by multiplying from the left by the matrix representing the first operation (in this case, a rotation) and then multiplying from the left again by the matrix representing the second operation (in this case, a reflection).

We then have the following results:

• A rotation followed by a rotation is a rotation: $Q_\varphi Q_\theta = Q_{\varphi+\theta}$
• A reflection followed by a reflection is a rotation: $H_\varphi Q_\theta = Q_{2(\varphi - \theta)}$
• A rotation followed by a reflection is a reflection: $H_\varphi Q_\theta = H_{\varphi-\frac{1}{2}\theta}$
• A reflection followed by a rotation is a reflection: $Q_\theta H_\varphi = H_{\varphi+\frac{1}{2}\theta}$

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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