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

\(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\), first rotates points through \( - \frac{{3\pi }}{4}\) radian (clockwise) and then reflects points through the horizontal \({x_1}\)-axis. [Hint: \(T\left( {{e_1}} \right) = \left( { - \frac{1}{{\sqrt 2 }},\frac{1}{{\sqrt 2 }}} \right)\)]

Usinglinear transformation,

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

When \({e_1}\) is rotated through \( - \frac{{3\pi }}{4}\) radian clockwise, then

\({e_1} \to \left[ {\begin{array}{*{20}{c}}{ - \frac{1}{{\sqrt 2 }}}\\{ - \frac{1}{{\sqrt 2 }}}\end{array}} \right]\).

And when reflected through the horizontal axis,

\({e_1} \to \left[ {\begin{array}{*{20}{c}}{ - \frac{1}{{\sqrt 2 }}}\\{ - \frac{1}{{\sqrt 2 }}}\end{array}} \right] \to \left[ {\begin{array}{*{20}{c}}{ - \frac{1}{{\sqrt 2 }}}\\{\frac{1}{{\sqrt 2 }}}\end{array}} \right]\).

When \({e_2}\) is rotated through \( - \frac{{3\pi }}{4}\) radian clockwise, then

\({e_2} \to \left[ {\begin{array}{*{20}{c}}{\frac{1}{{\sqrt 2 }}}\\{ - \frac{1}{{\sqrt 2 }}}\end{array}} \right]\).

And when reflected through the horizontal axis,

\({e_2} \to \left[ {\begin{array}{*{20}{c}}{\frac{1}{{\sqrt 2 }}}\\{ - \frac{1}{{\sqrt 2 }}}\end{array}} \right] \to \left[ {\begin{array}{*{20}{c}}{\frac{1}{{\sqrt 2 }}}\\{\frac{1}{{\sqrt 2 }}}\end{array}} \right]\).

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

\(T = \left[ {\begin{array}{*{20}{c}}{ - \frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\\{\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\end{array}} \right]x\).

So, the linear transformation matrixis \(\left[ {\begin{array}{*{20}{c}}{ - \frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\\{\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\end{array}} \right]\).