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Usinglinear transformation,

\(\begin{aligned}T &= T\left( {{x_1}{e_1} + {x_2}{e_2}} \right)\\ &= {x_1}T\left( {{e_1}} \right) + {x_2}T\left( {{e_2}} \right)\\ &= \left[ {\begin{array}{*{20}{c}}{T\left( {{e_1}} \right)}&{T\left( {{e_2}} \right)}\end{array}} \right]x\end{aligned}\)

For \({e_1}\), when it is reflected through the horizontal \({x_2}\)-axis, then

\( {e_1} \to - {e_1}\).

When it rotates points \(\frac{\pi }{2}\) radians,

\( - {e_1} \to - {e_2}\).

For \({e_1}\), when it is reflected through the horizontal \({x_1}\)-axis, then

\({e_2} \to - {e_2}\).

When it rotates points \(\frac{\pi }{2}\) radians,

\( - {e_2} \to - {e_1}\).

By the equation \(T = \left[ {\begin{array}{*{20}{c}}{T\left( {{e_1}} \right)}&{T\left( {{e_2}} \right)}\end{array}} \right]x\),

\(\begin{aligned}T &= \left[ {\begin{array}{*{20}{c}}{ - {e_2}}&{ - {e_1}}\end{array}} \right]x\\ &= \left[ {\begin{array}{*{20}{c}}0&{ - 1}\\{ - 1}&0\end{array}} \right]x\end{aligned}\)

By the equation \(T = \left[ {\begin{array}{*{20}{c}}0&{ - 1}\\{ - 1}&0\end{array}} \right]x\),the matrix\(A = \left[ {\begin{array}{*{20}{c}}0&{ - 1}\\{ - 1}&0\end{array}} \right]\).

So, the linear transformation matrix is \(\left[ {\begin{array}{*{20}{c}}0&{ - 1}\\{ - 1}&0\end{array}} \right]\).