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

\(T\left( x \right) = \left[ {\begin{array}{*{20}{c}}{0.5}&0\\0&{0.5}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\)

Short answer

The transformation \(T\left( x \right)\) represents the contraction by the factor \(0.5\).

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 thetransformation\(T\left( x \right) = \left[ {\begin{array}{*{20}{c}}{0.5}&0\\0&{0.5}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\).

\(\begin{aligned} T\left( x \right) &= \left[ {\begin{array}{*{20}{c}}{0.5}&0\\0&{0.5}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}5\\2\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{0.5 \times 5 + 0 \times 2}\\{0 \times 5 + 0.5 \times 2}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{2.5}\\1\end{array}} \right]\end{aligned}\)

Finding the rectangular coordinate

For the rectangular coordinate \(v = \left[ {\begin{array}{*{20}{c}}{ - 2}\\4\end{array}} \right]\), find the coordinate after thetransformation \(T\left( x \right) = \left[ {\begin{array}{*{20}{c}}{0.5}&0\\0&{0.5}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\).

\(\begin{aligned} T\left( x \right) &= \left[ {\begin{array}{*{20}{c}}{0.5}&0\\0&{0.5}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 2}\\4\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{0.5 \times \left( { - 2} \right) + 0 \times 4}\\{0 \times \left( { - 2} \right) + 0.5 \times 4}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{ - 1}\\2\end{array}} \right]\end{aligned}\)

Finding the rectangular coordinate

The transformed coordinates \(\left[ {\begin{array}{*{20}{c}}{2.5}\\1\end{array}} \right]\) and \(\left[ {\begin{array}{*{20}{c}}{ - 1}\\2\end{array}} \right]\) can be plotted as follows:

So, the transformation \(T\left( x \right)\) represents contraction by the factor \(0.5\).