Question

Let \({\bf{T}}:{\mathbb{R}^{\bf{2}}} \to {\mathbb{R}^{\bf{2}}}\) be the linear transformation that reflects each point through the \({{\bf{x}}_{\bf{1}}}\)-axis. That is, \({\bf{T}}\left( {\bf{x}} \right) = \left[ {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{0}}\\{\bf{0}}&{ - {\bf{1}}}\end{array}} \right]{\bf{x}}\). Make two sketches that illustrate properties (i) \({\bf{T}}\left( {u + {\bf{v}}} \right) = {\bf{T}}\left( {\bf{u}} \right) + {\bf{T}}\left( {\bf{v}} \right)\) and (ii) \({\bf{T}}\left( {{\bf{cw}}} \right) = {\bf{cT}}\left( {\bf{w}} \right)\)of a linear transformation.

Short answer

i)

ii)

Step-by-step solution

Determine the first property

Let \(u = \left[ {\begin{array}{*{20}{c}}1\\2\end{array}} \right],v = \left[ {\begin{array}{*{20}{c}}2\\1\end{array}} \right] \in {\mathbb{R}^2}\). Then,

\(\begin{array}{c}T\left( u \right) = T\left( {\left[ {\begin{array}{*{20}{c}}1\\2\end{array}} \right]} \right)\\ = \left[ {\begin{array}{*{20}{c}}1&0\\0&{ - 1}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1\\2\end{array}} \right]\\T\left( u \right) = \left[ {\begin{array}{*{20}{c}}1\\{ - 2}\end{array}} \right]\end{array}\)

\(\begin{array}{c}T\left( v \right) = T\left( {\left[ {\begin{array}{*{20}{c}}2\\1\end{array}} \right]} \right)\\ = \left[ {\begin{array}{*{20}{c}}1&0\\0&{ - 1}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}2\\1\end{array}} \right]\\T\left( v \right) = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\end{array}\)

And \(u + v = \left[ {\begin{array}{*{20}{c}}1\\2\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}2\\1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}3\\3\end{array}} \right]\). Therefore,

\(\begin{array}{c}T\left( {u + v} \right) = T\left( {\left[ {\begin{array}{*{20}{c}}3\\3\end{array}} \right]} \right)\\ = \left[ {\begin{array}{*{20}{c}}1&0\\0&{ - 1}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}3\\3\end{array}} \right]\\T\left( {u + v} \right) = \left[ {\begin{array}{*{20}{c}}3\\{ - 3}\end{array}} \right]\end{array}\)

Provide the image that illustrates property (i)

Note that \(T\) reflects about the \({x_1}\)-axis.

Determine the second property

Let \(c = 3 \in \mathbb{R}\) and \(w = \left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right] \in {\mathbb{R}^2}\). Then,

\(cw = 3\left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}3\\3\end{array}} \right]\)

Its image is given by:

\(\begin{array}{c}T\left( {cw} \right) = T\left( {\left[ {\begin{array}{*{20}{c}}3\\3\end{array}} \right]} \right)\\ = \left[ {\begin{array}{*{20}{c}}1&0\\0&{ - 1}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}3\\3\end{array}} \right]\\T\left( {cw} \right) = \left[ {\begin{array}{*{20}{c}}3\\{ - 3}\end{array}} \right]\end{array}\)

Provide the image that illustrates property (ii)

Note that \(T\) reflects about the \({x_1}\)-axis.