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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