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Magazine Chapter 1: Q16E (page 1)
In Exercises 13-16, use a rectangular coordinator system to plot \(u = \left[ {\begin{array}{*{20}{c}}5\\2\end{array}} \right]\), \(v = \left[ {\begin{array}{*{20}{c}}{ - 2}\\4\end{array}} \right]\) and their images under the given transformation \(T\). (Make a separate and reasonably large sketch for each exercise.) Describe geometrically what \(T\) does to each vector \(x\) in \({\mathbb{R}^2}\).
\(T\left( x \right) = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\)
Short Answer
The transformation \(T\left( x \right)\) represents the projection on the \({x_1} = {x_2}\) axis.
Step by step solution
Finding the rectangular coordinate
For therectangular coordinate \(u = \left[ {\begin{array}{*{20}{c}}5\\2\end{array}} \right]\), find the coordinate after the transformation \(T\left( x \right) = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\).
\(\begin{aligned} T\left( x \right) &= \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]\left[ {\begin{array}{*{20}{c}}5\\2\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{0 \times 5 + 1 \times 2}\\{1 \times 5 + 0 \times 2}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}2\\5\end{array}} \right]\end{aligned}\)
Finding the rectangular coordinate
For therectangular coordinate \(v = \left[ {\begin{array}{*{20}{c}}{ - 2}\\4\end{array}} \right]\), find the coordinate after the transformation \(T\left( x \right) = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\).
\(\begin{aligned} T\left( x \right) &= \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 2}\\4\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{0 \times \left( { - 2} \right) + 1 \times 4}\\{1 \times \left( { - 2} \right) + 0 \times 4}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}4\\{ - 2}\end{array}} \right]\end{aligned}\)
Finding the rectangular coordinate
The transformed coordinates \(\left[ {\begin{array}{*{20}{c}}2\\5\end{array}} \right]\) and \(\left[ {\begin{array}{*{20}{c}}4\\{ - 2}\end{array}} \right]\) can be plotted as follows:
So, the transformation \(T\left( x \right)\) represents the reflection through the line \({x_2} = {x_1}\).
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