Question

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 \({\mathbb{R}^2}\).

\(T\left( x \right) = \left[ {\begin{array}{*{20}{c}}{ - 1}&0\\0&{ - 1}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\)

Short answer

The transformation \(T\left( x \right)\) can be defined as the reflection about the origin or the rotation about the origin either by \(\pi \) or \( - \pi \) radian.

Step-by-step solution

Finding the rectangular coordinate

For the rectangular coordinate \(u = \left[ {\begin{array}{*{20}{c}}5\\2\end{array}} \right]\), find the coordinate after thetransformation \(T\left( x \right) = \left[ {\begin{array}{*{20}{c}}{ - 1}&0\\0&{ - 1}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\).

\(\begin{aligned} T\left( x \right) &= \left[ {\begin{array}{*{20}{c}}{ - 1}&0\\0&{ - 1}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}5\\2\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{\left( { - 1} \right) \times 5 + 0 \times 2}\\{0 \times 5 + \left( { - 1} \right) \times 2}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{ - 5}\\{ - 2}\end{array}} \right]\end{aligned}\)

Finding the rectangular coordinate

For the rectangular coordinate \(v = \left[ {\begin{array}{*{20}{c}}{ - 2}\\4\end{array}} \right]\), find the coordinate after thetransformation \(T\left( x \right) = \left[ {\begin{array}{*{20}{c}}{ - 1}&0\\0&{ - 1}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\).

\(\begin{aligned} T\left( x \right) &= \left[ {\begin{array}{*{20}{c}}{ - 1}&0\\0&{ - 1}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 2}\\4\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{\left( { - 1} \right) \times \left( { - 2} \right) + 0 \times 4}\\{0 \times \left( { - 2} \right) + \left( { - 1} \right) \times 4}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}2\\{ - 4}\end{array}} \right]\end{aligned}\)

Finding the rectangular coordinate

The transformed coordinates can be plotted as follows:

So, the transformation \(T\left( x \right)\) can be defined as the reflection about the origin or the rotation about the origin either by \(\pi \) or \( - \pi \) radian.