Question

Consider a unit vector $\stackrel{\mathbf{\perp }}{\mathbf{u}}\mathbf{}\mathbf{in}\mathbf{}{\mathit{R}}^{\mathbf{3}}$. We define the matrices

$\mathit{A}\mathbf{=}\mathbf{2}\stackrel{\mathbf{\perp }}{\mathbf{u}}\stackrel{\mathbf{\perp }}{{\mathbf{u}}^{\mathbf{T}}}\mathbf{}\mathbf{}\mathbf{-}{\mathit{I}}_{\mathbf{3}}\mathbf{}\mathbf{and}\mathbf{}\mathit{B}\mathbf{-}{\mathit{I}}_{\mathbf{3}}\mathbf{-}\mathbf{2}\stackrel{\mathbf{\perp }}{\mathbf{u}}\stackrel{\mathbf{\perp }}{{\mathbf{u}}^{\mathbf{T}}}$

Describe the linear transformations defined by these matrices geometrically.

Short answer

A is the reflection of $\stackrel{\perp }{x}$ about line L that is spanned by $\stackrel{\perp }{u}$.

B is the reflection of $\stackrel{\perp }{x}$ about line V that is spanned by $\stackrel{\perp }{u}$.

Step-by-step solution

Reflection of x about line L.

Let the line L be spanned by a unit vector $\stackrel{\perp }{u}\mathrm{in}{R}^3$

By using the definition 2.2.2 the reflection of $\stackrel{\perp }{x}$about L is,

$$\begin{gathered} {\mathrm{ref}}_{\mathrm{L}}=2\mathrm{proj}\left(\stackrel{\perp }{\mathrm{x}}\right)-\stackrel{\perp }{\mathrm{x}} \\ =2\left(\stackrel{\mathrm{r}}{\mathrm{x}}.\stackrel{\mathrm{r}}{\mathrm{u}}\right)\stackrel{\mathrm{r}}{\mathrm{u}}-\stackrel{\mathrm{r}}{\mathrm{x}} \\ =2\stackrel{\mathrm{r}}{\mathrm{u}}\stackrel{\mathrm{r}}{{\mathrm{u}}^{\mathrm{T}}\stackrel{\mathrm{r}}{\mathrm{x}}-\stackrel{\mathrm{r}}{\mathrm{x}}} \\ =\left(2\stackrel{\mathrm{r}}{\mathrm{u}}\stackrel{\mathrm{r}}{{\mathrm{u}}^{\mathrm{T}}-{\mathrm{I}}_3}\right)\stackrel{\mathrm{r}}{\mathrm{x}} \\ ={\mathrm{A}}_{\mathrm{x}}^{\mathrm{r}} \end{gathered}$$

Reflection of x⊥ about line V.

Let Va plane with a normal vector $\stackrel{\perp }{u}$

Then the reflection of$\stackrel{\perp }{\mathrm{x}}$ about line V.

$$\begin{gathered} re{f}_v=pro{j}_v\left(\stackrel{\perp }{X}\right)-pro{j}_L\stackrel{\perp }{X} \\ =\left(\left(\stackrel{r}{X}\right)-pro{j}_L\stackrel{r}{X}\right)-pro{j}_L\stackrel{r}{X} \\ =\stackrel{r}{X}-2pro{j}_L\stackrel{r}{X} \\ =\stackrel{r}{X}-2\left(\stackrel{r}{X}.\stackrel{r}{X}\right)\stackrel{r}{X} \\ =\stackrel{r}{X}-2\stackrel{r}{u}\stackrel{r}{u^T\stackrel{r}{X}} \\ =\left(I_3-2\stackrel{r}{u}\stackrel{r}{u^T}\right)\stackrel{r}{X} \\ =B_x^r \end{gathered}$$

Hence, the answer is

A will be the reflection of $\stackrel{\perp }{x}$ about line L that is spanned by $\stackrel{\perp }{u}$.

B will be the reflection of $\stackrel{\perp }{x}$ about line V that is spanned by $\stackrel{\perp }{u}$.