Let \(x = \left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\), \({v_1} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\5\end{array}} \right]\) and \({v_2} = \left[ {\begin{array}{*{20}{c}}7\\{ - 3}\end{array}} \right]\) and let \(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\) be a linear transformation that maps \(x\) into \({x_1}{v_1} + {x_1}{v_2}\). Find a matrix \(A\) such that \(T\left( x \right)\) is \(Ax\) for each \(x\).

\(\begin{aligned}T\left( x \right) &= {x_1}{v_1} + {x_2}{v_2}\\ &= \left[ {\begin{array}{*{20}{c}}{{v_1}}&{{v_2}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\end{aligned}\)

Substitute the values of \({v_1}\) and \({v_2}\) in the equation \(T\left( x \right) = \left[ {\begin{array}{*{20}{c}}{{v_1}}&{{v_2}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\).

\(\begin{aligned}{c}T\left( x \right) &= \left[ {\begin{array}{*{20}{c}}{{v_1}}&{{v_2}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{ - 2}&7\\5&{ - 3}\end{array}} \right]x\end{aligned}\)

Compare the equation \(T\left( x \right) = \left[ {\begin{array}{*{20}{c}}{ - 2}&7\\5&{ - 3}\end{array}} \right]x\)with\(T\left( x \right) = Ax\).

\(A = \left[ {\begin{array}{*{20}{c}}{ - 2}&7\\5&{ - 3}\end{array}} \right]\)

So, matrix \(A\) is \(\left[ {\begin{array}{*{20}{c}}{ - 2}&7\\5&{ - 3}\end{array}} \right]\).