Question

Let \(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\) be the linear transformation with standard matrix \(A = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}\end{array}} \right]\), where \({a_1}\) and \({a_2}\) are shown in the figure. Using the figure, draw the image of \(\left[ {\begin{array}{*{20}{c}}{ - 1}\\3\end{array}} \right]\) under the transformation \(T\).

Short answer

Solve the equation \(T\left( x \right) = Ax\) using thelinear transformation.

\(\begin{aligned} T\left( x \right) &= Ax\\ &= \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}\end{array}} \right]x\\ &= {x_1}{a_1} + {x_2}{a_2}\end{aligned}\)

Step-by-step solution

Solve the equation \(T\left( x \right) = Ax\)

Solve the equation \(T\left( x \right) = Ax\) using thelinear transformation.

\(\begin{aligned} T\left( x \right) &= Ax\\ &= \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}\end{array}} \right]x\\ &= {x_1}{a_1} + {x_2}{a_2}\end{aligned}\)

Find the image of \(\left[ {\begin{array}{*{20}{c}}{ - 1}\\3\end{array}} \right]\)

Find the image of \(\left[ {\begin{array}{*{20}{c}}{ - 1}\\3\end{array}} \right]\) using the equation \(T\left( x \right) = {x_1}{a_1} + {x_2}{a_2}\).

\(T\left( x \right) = - {a_1} + 3{a_2}\)

Locate the image of \(\left( { - 1,3} \right)\).

In the given graph, locate the image of \(\left( { - 1,3} \right)\) by forming a parallelogram as shown below: