Step 1
Given Vectors:
\[{\bf{x}} = \left[ {\begin{array}{*{20}{l}}{{x_1}}\\{{x_2}}\end{array}} \right],\,\,{{\bf{v}}_1} = \left[ {\begin{array}{*{20}{c}}{ - 3}\\5\end{array}} \right],{\rm{ and }}\,\,{{\bf{v}}_2} = \left[ {\begin{array}{*{20}{c}}7\\{ - 2}\end{array}} \right]\]Given Transformation:\[T:{R^2} \to {R^2}::\,\,T\left( {\bf{x}} \right) = {x_1}{{\bf{v}}_{\bf{1}}} + {x_2}{{\bf{v}}_2}\]We have to find a matrix A, such that \[T\left( {\bf{x}} \right) = A{\bf{x}}\]

Step 2: Substitute the vectors to find the transformation
\[\begin{array}{l}\,T\left( {\bf{x}} \right) = {x_1}{{\bf{v}}_{\bf{1}}} + {x_2}{{\bf{v}}_2}\\\\ = {x_1}\left[ {\begin{array}{*{20}{c}}{ - 3}\\5\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}7\\{ - 2}\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}{ - 3{x_1} + 7{x_2}}\\{5{x_1} - 2{x_2}}\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}{ - 3}&7\\5&{ - 2}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\end{array}\]So, the required matrix A is given by the coefficient matrix of the vector $\bf{x}$. \[\begin{array}{l}\,T\left( {\bf{x}} \right) = \left[ {\begin{array}{*{20}{c}}{ - 3}&7\\5&{ - 2}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\\\\\,T\left( {\bf{x}} \right) = A{\bf{x}}\end{array}\]

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