Question

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

\(T:{\mathbb{R}^3} \to {\mathbb{R}^2}\), rotates points (about the origin) through\( - \frac{\pi }{4}\)radians (clockwise).[Hint:\(T\left( {{e_1}} \right) = \left[ {\frac{1}{{\sqrt 2 }}, - \frac{1}{{\sqrt 2 }}} \right]\)]

Short answer

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

Step-by-step solution

Find the value of \(T\) using linear transformation

Usinglinear transformation,

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

Find the transformation for \(T\left( {{e_1}} \right)\) and \(T\left( {{e_2}} \right)\)

Transformation represents the rotation of \( - \frac{\pi }{4}\) radian about the origin (counterclockwise).

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

And

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

Find the transformation for \(T\left( {{e_1}} \right)\) and \(T\left( {{e_2}} \right)\)

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

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

Find the standard matrix \(T\) for linear transformation

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

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