In Exercise 1-10, assume that is a linear transformation. Find the standard matrix of.\(T\)

\(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\), firsts reflects points through the horizontal \({x_1}\)-axis and then reflects point through the line \({x_2} = {x_1}\).

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 \({x_1}\)-axis, then

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

When it is reflected through the line \({x_2} = {x_1}\),

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

For \({e_2}\), when it is reflected through the horizontal \({x_1}\)-axis, then

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

When it is reflected through the line \({x_2} = {x_1}\),

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

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