Question

Let \(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\) be a linear transformation such that \(T\left( {{x_1},{x_2}} \right) = \left( {{x_1} + {x_2},4{x_1} + 5{x_2}} \right)\). Find \({\mathop{\rm x}\nolimits} \) such that \(T\left( x \right) = \left( {3,8} \right)\).

Short answer

The value of \(x\) is \(x = \left[ {\begin{array}{*{20}{c}}7\\{ - 4}\end{array}} \right]\) such that \(T\left( x \right) = \left( {3,8} \right)\).

Step-by-step solution

Determine the standard matrix of \(T\) by inspection

It is given that \(T\left( x \right) = \left[ {\begin{array}{*{20}{c}}3\\8\end{array}} \right]\).

Write the linear transformation into the standard matrix of \(T\) by inspection.

\[\begin{array}{c}T\left( x \right) = \left[ {\begin{array}{*{20}{c}}{{x_1} + {x_2}}\\{4{x_1} + 5{x_2}}\end{array}} \right]\\\left[ {\begin{array}{*{20}{c}}3\\8\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\\\left[ {\begin{array}{*{20}{c}}3\\8\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&1\\4&5\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\end{array}\]

Write the standard matrix into an augmented matrix

Write the standard matrix into an augmented matrix.

\(\left[ {\begin{array}{*{20}{c}}1&1&3\\4&5&8\end{array}} \right]\)

Apply row operation

At row 2, multiply row 1 by 4 and subtract it from row 2.

\(\left[ {\begin{array}{*{20}{c}}1&1&3\\0&1&{ - 4}\end{array}} \right]\)

Apply row operation to find \(x\)

At row 1, multiply row 2 by \(1\) and subtract it from row 1.

\(\left[ {\begin{array}{*{20}{c}}1&0&7\\0&1&{ - 4}\end{array}} \right]\)

Thus, the value of \(x\) is \(x = \left[ {\begin{array}{*{20}{c}}7\\{ - 4}\end{array}} \right]\) such that \(T\left( x \right) = \left( {3,8} \right)\).