Let \(T:{\mathbb{R}^3} \to {\mathbb{R}^3}\) be the linear transformation that reflects each vector through the plane \({x_{\bf{2}}} = 0\). That is, \(T\left( {{x_1},{x_2},{x_3}} \right) = \left( {{x_1}, - {x_2},{x_3}} \right)\). Find the standard matrix of \(T\).

Use the equation \(T = A{\bf{x}}\). As the order of matrix \(T\) is \(3 \times 1\) and the order of \({\bf{x}}\) is \(3 \times 1\), the order of \(A\) must be \(3 \times 3\).

\(\left( {\begin{aligned}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{ - {x_2}}\\{{x_3}}\end{aligned}} \right)\)

Compare both sides of the equation \(\left( {\begin{aligned}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{ - {x_2}}\\{{x_3}}\end{aligned}} \right)\) to get the first row of the matrix with unknown elements as \(\left( {\begin{aligned}{*{20}{c}}1&0&0\end{aligned}} \right)\).

Compare both sides of the equation \(\left( {\begin{aligned}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{ - {x_2}}\\{{x_3}}\end{aligned}} \right)\) to get the second row of the matrix with unknown elements as \(\left( {\begin{aligned}{*{20}{c}}0&{ - 1}&0\end{aligned}} \right)\).

Compare both sides of the equation \(\left( {\begin{aligned}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{ - {x_2}}\\{{x_3}}\end{aligned}} \right)\) to get the third row of the matrix with unknown elements as \(\left( {\begin{aligned}{*{20}{c}}0&0&1\end{aligned}} \right)\).

So, the unknown matrix in the equation is \(\left( {\begin{aligned}{*{20}{c}}1&0&0\\0&{ - 1}&0\\0&0&1\end{aligned}} \right)\).