Question

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

$\mathit{A}\mathbf{=}\mathbf{2}\stackrel{\mathbf{\perp }}{\mathbf{u}}\stackrel{\mathbf{\perp }}{\mathbf{u}}\mathbf{}\mathbf{-}{\mathit{l}}_{\mathbf{3}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{B}\mathbf{=}{\mathit{l}}_{\mathbf{3}}\mathbf{-}\mathbf{2}\stackrel{\mathbf{\perp }}{\mathbf{u}}\stackrel{\mathbf{\perp }}{\mathbf{u}}\mathbf{}$

Describe the linear transformations defined by these matrices geometrically

Short answer

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

B is the reflection of $\stackrel{\perp }{\mathrm{x}}$ 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 $\stackrel{\perp }{u}$in $R^3$

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

$$\begin{gathered} {\mathrm{ref}}_{\mathrm{L}}=2\mathrm{proj}\left(\stackrel{\perp }{\mathrm{x}}\right)-\stackrel{\perp }{\mathrm{x}} \\ =2\left(\stackrel{\mathrm{r}}{\mathrm{x}}.\stackrel{\mathrm{r}}{\mathrm{u}}\right)\stackrel{\mathrm{r}}{\mathrm{u}}-\stackrel{\mathrm{r}}{\mathrm{x}} \\ =2\stackrel{\mathrm{r}}{\mathrm{u}}\stackrel{\mathrm{r}}{\mathrm{u}}{}^{\mathrm{T}}\stackrel{\mathrm{r}}{\mathrm{x}}-\stackrel{\mathrm{r}}{\mathrm{x}} \\ =\left(2\stackrel{r}{u}\stackrel{r}{u}{}^T-I_3\right)\stackrel{\mathrm{r}}{\mathrm{x}} \\ =\mathrm{A}\stackrel{\mathrm{r}}{\mathrm{x}} \end{gathered}$$

Reflection of x⊥ about line V.

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

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

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

Hence, the answer is

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

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