Question

In Exercises 33 and 34, Tis a linear transformation from \({\mathbb{R}^2}\) into \({\mathbb{R}^2}\). Show that T is invertible and find a formula for \({T^{ - 1}}\).

34. \(T\left( {{x_1},{x_2}} \right) = \left( {6{x_1} - 8{x_2}, - 5{x_1} + 7{x_2}} \right)\)

Short answer

The formula for \({T^{ - 1}}\) is \({T^{ - 1}}\left( {{x_1},{x_2}} \right) = \left( {\frac{7}{2}{x_1} + 4{x_2},\frac{5}{2}{x_1} + 3{x_2}} \right)\).

Step-by-step solution

Determine the standard matrix T

Write the transformation \(T\left( x \right)\) and \(x\) in the column vector of \(A\).

\(\begin{aligned}{c}T\left( x \right) = \left( {\begin{aligned}{*{20}{c}}{6{x_1} - 8{x_2}}\\{ - 5{x_1} + 7{x_3}}\end{aligned}} \right)\\ = \left( A \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}6&{ - 8}\\{ - 5}&7\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\end{aligned}} \right)\end{aligned}\)

Thus, the standard matrix of T is \(A = \left( {\begin{aligned}{*{20}{c}}6&{ - 8}\\{ - 5}&7\end{aligned}} \right)\).

Show that T  is invertible

Theorem 4states that \(A = \left( {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right)\). If \(ad - bc \ne 0\), then A is invertible.

\({A^{ - 1}} = \frac{1}{{ad - bc}}\left( {\begin{aligned}{*{20}{c}}d&{ - b}\\{ - c}&a\end{aligned}} \right)\)

If \(ad - bc = 0\), then Aisnot invertible.

The linear transformation T is invertible since det\(A = 2 \ne 0\).

Determine the formula for \({T^{ - 1}}\)

Let\(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be a linear transformation and Abe the standard matrix for T. Then, according toTheorem 9,Tis invertible if and only if Ais an invertible matrix. The linear transformation S, given by \(S\left( x \right) = {A^{ - 1}}{\mathop{\rm x}\nolimits} \), is a unique function satisfying the equations

1. \(S\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \)for all x in \({\mathbb{R}^n}\), and
2. \(T\left( {S\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \)for all x in \({\mathbb{R}^n}\).

According to theorem 9, transformation Tis invertible and \({T^{ - 1}}\left( x \right) = Bx\), where\(B = {A^{ - 1}}\).

Use the formula for \(2 \times 2\) inverse.

\(\begin{aligned}{c}{A^{ - 1}} = \frac{1}{{42 - 40}}\left( {\begin{aligned}{*{20}{c}}7&8\\5&6\end{aligned}} \right)\\ = \frac{1}{2}\left( {\begin{aligned}{*{20}{c}}7&8\\5&6\end{aligned}} \right)\end{aligned}\)

Therefore,

\(\begin{aligned}{c}{T^{ - 1}}\left( {{x_1},{x_2}} \right) = {A^{ - 1}}x\\ = \frac{1}{2}\left( {\begin{aligned}{*{20}{c}}7&8\\5&6\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\end{aligned}} \right)\\ = \left( {\frac{7}{2}{x_1} + 4{x_2},\frac{5}{2}{x_1} + 3{x_2}} \right)\end{aligned}\)

Thus, the formula for \({T^{ - 1}}\) is \({T^{ - 1}}\left( {{x_1},{x_2}} \right) = \left( {\frac{7}{2}{x_1} + 4{x_2},\frac{5}{2}{x_1} + 3{x_2}} \right)\).