91影视

Let us consider$x_1-2x_2+2x_3=0$is an equation of a plane.

Also let us consider the basis of the plane consists of two vectors parallel to the plane and one vector perpendicular to the plane.

i.e.$I=\left\{u鈨�,v鈨�,w鈨�\right\}$is the basis of the plane such thatlocalid="1659354468607" $\stackrel{鈬赌}{U},\stackrel{鈬赌}{V}$are parallel to the plane and is perpendicular to the plane. Then we have $\stackrel{鈬赌}{w}=\stackrel{鈬赌}{U}脳\stackrel{鈬赌}{V}$.

Since$I=\left\{u鈨�,v鈨�,w鈨�\right\}$ is the basis of the plane such thatrole="math" localid="1659354529564" $\stackrel{鈬赌}{U},\stackrel{鈬赌}{V}$ are parallel to the plane androle="math" localid="1659354504549" $\stackrel{鈬赌}{w}$ is perpendicular to the plane, then the linear transformation of the vectors $\stackrel{鈬赌}{U},\stackrel{鈬赌}{V}$and$\stackrel{鈬赌}{w}$ is given by

role="math" localid="1659354641352" $$\begin{gathered} T\left(\stackrel{鈬赌}{u}\right)=\stackrel{鈬赌}{u} \\ T(v鈨�)=\stackrel{鈬赌}{v} \\ T(\stackrel{鈬赌}{w}鈨�)=-\stackrel{鈬赌}{w} \end{gathered}$$

Therefore the matrix $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$which is clearly similar to the given matrix .

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$

If a 3 x 3 matrix A represents the reflection about a plane, then A is similar to the matrix $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$