Question

Let $\mathbf{T}\left(\overrightarrow{x}\right)\mathbf{=}{\mathbf{ref}}_{\mathbf{L}}\left(\overrightarrow{x}\right)$be the reflection about the line L in${\mathrm{ℝ}}^2$shown in the accompanying figure.

a. Draw sketches to illustrate that T is linear.

b. Find the matrix of T in terms of$\mathit{\theta }$.

Short answer

a. The graph is,

b. The matrix is,

$\mathrm{T}\left(\overrightarrow{\mathrm{x}}\right)=\left[\begin{array}{cc}\cos 2\mathrm{\theta }& \sin 2\mathrm{\theta }\\ \sin 2\mathrm{\theta }& -\cos 2\mathrm{\theta }\end{array}\right]\left(\overrightarrow{\mathrm{x}}\right)$

Step-by-step solution

Draw the sketch

The graph of the line L is,

The reflection of vector $\overrightarrow{\mathrm{x}}$about the line L in ${\mathrm{ℝ}}^2$is,

${\mathrm{ref}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right)=2{\mathrm{proj}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right)-\left(\overrightarrow{\mathrm{x}}\right)$

Therefore, the graph of the line T is,

Compute the projection.

The projection of a vector $\overrightarrow{\mathrm{x}}$onto $\overrightarrow{\mathrm{u}}$is,

$$\begin{gathered} {\mathrm{proj}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right)=\left(\frac{\overrightarrow{\mathrm{x}}·\overrightarrow{\mathrm{u}}}{\mathrm{uu}}\right)\overrightarrow{\mathrm{u}} \\ \mathrm{Here}, \\ \overrightarrow{\mathrm{x}}=\left[\begin{array}{c}{\mathrm{x}}_1\\ {\mathrm{x}}_2\end{array}\right],\overrightarrow{\mathrm{u}}=\left[\begin{array}{c}\mathrm{cos\theta }\\ \mathrm{sin\theta }\end{array}\right] \\ \mathrm{Thus},\mathrm{the}\mathrm{projection}\mathrm{is}, \\ {\mathrm{proj}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right)=\left(\begin{array}{c}\left[\begin{array}{c}{\mathrm{x}}_1\\ {\mathrm{x}}_2\end{array}\right]·\left[\begin{array}{c}\mathrm{cos\theta }\\ \mathrm{sin\theta }\end{array}\right]\\ \left[\begin{array}{c}\mathrm{cos\theta }\\ \mathrm{sin\theta }\end{array}\right]·\left[\begin{array}{c}\mathrm{cos\theta }\\ \mathrm{sin\theta }\end{array}\right]\end{array}\right)\left[\begin{array}{c}\mathrm{cos\theta }\\ \mathrm{sin\theta }\end{array}\right] \\ {\mathrm{proj}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right)=\left(\frac{{\mathrm{cos\theta x}}_1+{\mathrm{sin\theta x}}_2}{{\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta }}\right)\left[\begin{array}{c}\mathrm{cos\theta }\\ \mathrm{sin\theta }\end{array}\right] \\ {\mathrm{proj}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right)=\left({\mathrm{cos\theta x}}_1+{\mathrm{sin\theta x}}_2\right)\left[\begin{array}{c}\mathrm{cos\theta }\\ \mathrm{sin\theta }\end{array}\right] \\ {\mathrm{proj}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right)=\left[\begin{array}{c}\left({\mathrm{cos\theta x}}_1+{\mathrm{sin\theta x}}_2\right)\mathrm{cos\theta }\\ \left({\mathrm{cos\theta x}}_1+{\mathrm{sin\theta x}}_2\right)\mathrm{sin\theta }\end{array}\right] \\ {\mathrm{proj}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right)=\left[\begin{array}{c}\left({\cos }^2{\mathrm{\theta x}}_1+\mathrm{cos\theta }{\mathrm{sin\theta x}}_2\right)\\ \left(\mathrm{cos\theta sin\theta }{\mathrm{x}}_1+{\sin }^2\mathrm{\theta }\right)\end{array}\right] \\ {\mathrm{proj}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right)=\left[\begin{array}{cc}{\cos }^2\mathrm{\theta }& \mathrm{cos\theta sin\theta }\\ \mathrm{cos\theta sin\theta }& {\sin }^2\mathrm{\theta }\end{array}\right]\overrightarrow{\mathrm{x}} \end{gathered}$$

Compute the reflection

$$\begin{gathered} \mathrm{The}\mathrm{reflection}\mathrm{of}\mathrm{the}\mathrm{vector}\overrightarrow{\mathrm{x}}\mathrm{is}, \\ \mathrm{T}\left(\overrightarrow{\mathrm{x}}\right)={\mathrm{ref}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right) \\ \mathrm{Here},{\mathrm{ref}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right)=2{\mathrm{proj}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right)-\left(\overrightarrow{\mathrm{x}}\right) \\ {\mathrm{ref}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right)=2\left[\begin{array}{cc}{\cos }^2\mathrm{\theta }& \mathrm{cos\theta sin\theta }\\ \mathrm{cos\theta sin\theta }& {\sin }^2\mathrm{\theta }\end{array}\right]\left(\overrightarrow{\mathrm{x}}\right)-\left(\overrightarrow{\mathrm{x}}\right) \\ {\mathrm{ref}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right)=2\left(\left[\begin{array}{cc}{\cos }^2\mathrm{\theta }& \mathrm{cos\theta sin\theta }\\ \mathrm{cos\theta sin\theta }& {\sin }^2\mathrm{\theta }\end{array}\right]-1\right)\left(\overrightarrow{\mathrm{x}}\right) \\ {\mathrm{ref}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right)=2\left(\left[\begin{array}{cc}{\cos }^2\mathrm{\theta }& \mathrm{cos\theta sin\theta }\\ \mathrm{cos\theta sin\theta }& {\sin }^2\mathrm{\theta }\end{array}\right]-\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\right)\left(\overrightarrow{\mathrm{x}}\right) \\ {\mathrm{ref}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right)=\left(\left[\begin{array}{cc}2{\cos }^2\mathrm{\theta }-1& 2\mathrm{cos\theta sin\theta }\\ 2\mathrm{cos\theta sin\theta }& 2{\sin }^2\mathrm{\theta }-1\end{array}\right]\right)\left(\overrightarrow{\mathrm{x}}\right) \\ {\mathrm{ref}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right)=\left[\begin{array}{cc}\cos 2\mathrm{\theta }& \sin 2\mathrm{\theta }\\ \sin 2\mathrm{\theta }& -\cos 2\mathrm{\theta }\end{array}\right]\left(\overrightarrow{\mathrm{x}}\right) \end{gathered}$$

Compute the matrix

$$\begin{gathered} \mathrm{T}\left(\overrightarrow{\mathrm{x}}\right)={\mathrm{ref}}_{\mathrm{L}}\left(\overrightarrow{\mathrm{x}}\right) \\ \mathrm{T}\left(\overrightarrow{\mathrm{x}}\right)=\left[\begin{array}{cc}\cos 2\mathrm{\theta }& \sin 2\mathrm{\theta }\\ \sin 2\mathrm{\theta }& -\cos 2\mathrm{\theta }\end{array}\right]\left(\overrightarrow{\mathrm{x}}\right) \end{gathered}$$