91Ó°ÊÓ

The equation$A\stackrel{→}{\mathrm{Ï\dots }}=\stackrel{→}{\mathrm{Ï\dots }}$, implies$(A−I)\stackrel{→}{\mathrm{Ï\dots }}=0$, so, the homogeneous equations are,

$\begin{array}{l}(a−1){\mathrm{Ï\dots }}_1+b{\mathrm{Ï\dots }}_2=0\\ b{\mathrm{Ï\dots }}_1−(a+1){\mathrm{Ï\dots }}_2=0\end{array}$

The values of the components for $b≠0$will be,

${\mathrm{Ï\dots }}_1=\frac{b}{1−a}{\mathrm{Ï\dots }}_2,{\mathrm{Ï\dots }}_1=\frac{a+1}{b}{\mathrm{Ï\dots }}_2$

Consider only one value for the vector, so, role="math" localid="1664256441809" $\mathrm{Ï\dots }=\left[\begin{array}{l}b\\ 1−a\end{array}\right]$

The equation$A\stackrel{→}{w}=−\stackrel{→}{w}$, implies$(A+I)\stackrel{→}{w}=0$, so, the homogeneous equations are,

$\begin{array}{l}(a+1)w_1+b{w}_2=0\\ b{w}_1−(a−1)w_2=0\end{array}$

The values of the components for $b≠0$will be,

$w_1=\frac{−b}{1+a}{w}_2,w_1=\frac{a−1}{b}{w}_2$

Consider only one value for the vector, so, $w=\left[\begin{array}{l}−b\\ 1+a\end{array}\right]$.

The vector $\stackrel{→}{x}$is a composition of $x^{âˆ\yen }$and$x^{âŠ\yen }$.

That is,$\stackrel{→}{x}=x^{âˆ\yen }+x^{âŠ\yen }$

Where,$x^{âˆ\yen }=m\stackrel{→}{\mathrm{Ï\dots }},x^{âŠ\yen }=m\stackrel{→}{w}$

Consider the system.

$\begin{array}{c}T\left(\stackrel{→}{x}\right)=A\left(\stackrel{→}{x}\right)\\ ⇒T\left(\stackrel{→}{x}\right)=A(x^{âˆ\yen }+x^{âŠ\yen })\\ ⇒T\left(\stackrel{→}{x}\right)=A(m\stackrel{→}{\mathrm{Ï\dots }}+n\stackrel{→}{w})\\ ⇒T\left(\stackrel{→}{x}\right)=m\left(A\stackrel{→}{\mathrm{Ï\dots }}\right)+n\left(A\stackrel{→}{w}\right)\end{array}$

Here,$A\stackrel{→}{\mathrm{Ï\dots }}=\stackrel{→}{\mathrm{Ï\dots }},A\stackrel{→}{w}=−\stackrel{→}{w}$

Thus, the reflection is,

$\begin{array}{c}T\left(\stackrel{→}{x}\right)=m\left(\stackrel{→}{\mathrm{Ï\dots }}\right)+n(−\stackrel{→}{w})\\ ⇒T\left(\stackrel{→}{x}\right)=m\stackrel{→}{\mathrm{Ï\dots }}−n\stackrel{→}{w}\\ ⇒T\left(\stackrel{→}{x}\right)=x^{âˆ\yen }−x^{âŠ\yen }\\ ∴T\left(\stackrel{→}{x}\right)=re{f}_L\left(\stackrel{→}{\mathrm{Ï\dots }}\right)\end{array}$

Hence, the nonzero vectors are, $\mathrm{Ï\dots }=\left[\begin{array}{l}b\\ 1−a\end{array}\right]$and$w=\left[\begin{array}{l}−b\\ 1+a\end{array}\right]$.

As $T\left(\stackrel{→}{x}\right)=re{f}_L\left(\stackrel{→}{\mathrm{Ï\dots }}\right)$, hence it is proved that, $T\left(\stackrel{→}{x}\right)=A\left(\stackrel{→}{x}\right)$ represents the reflection about the line L spanned by$\stackrel{→}{\mathrm{Ï\dots }}$.