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Chapter 2: Q2.3-34Q (page 93)

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

Expert verified

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

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01

Determine the standard matrix T

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

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

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

02

Show that T  is invertible

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

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

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

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

03

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.

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

Therefore,

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

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

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Most popular questions from this chapter

Assume \(A - s{I_n}\) is invertible and view (8) as a system of two matrix equations. Solve the top equation for \({\bf{x}}\) and substitute into the bottom equation. The result is an equation of the form \(W\left( s \right){\bf{u}} = {\bf{y}}\), where \(W\left( s \right)\) is a matrix that depends upon \(s\). \(W\left( s \right)\) is called the transfer function of the system because it transforms the input \({\bf{u}}\) into the output \({\bf{y}}\). Find \(W\left( s \right)\) and describe how it is related to the partitioned system matrix on the left side of (8). See Exercise 15.

Let \(A = \left[ {\begin{array}{*{20}{c}}B&{\bf{0}}\\{\bf{0}}&C\end{array}} \right]\), where \(B\) and \(C\) are square. Show that \(A\)is invertible if an only if both \(B\) and \(C\) are invertible.

Describe in words what happens when you compute \({A^{\bf{5}}}\), \({A^{{\bf{10}}}}\), \({A^{{\bf{20}}}}\), and \({A^{{\bf{30}}}}\) for \(A = \left( {\begin{aligned}{*{20}{c}}{1/6}&{1/2}&{1/3}\\{1/2}&{1/4}&{1/4}\\{1/3}&{1/4}&{5/12}\end{aligned}} \right)\).

[M] Suppose memory or size restrictions prevent your matrix program from working with matrices having more than 32 rows and 32 columns, and suppose some project involves \(50 \times 50\) matrices A and B. Describe the commands or operations of your program that accomplish the following tasks.

a. Compute \(A + B\)

b. Compute \(AB\)

c. Solve \(Ax = b\) for some vector b in \({\mathbb{R}^{50}}\), assuming that \(A\) can be partitioned into a \(2 \times 2\) block matrix \(\left[ {{A_{ij}}} \right]\), with \({A_{11}}\) an invertible \(20 \times 20\) matrix, \({A_{22}}\) an invertible \(30 \times 30\) matrix, and \({A_{12}}\) a zero matrix. [Hint: Describe appropriate smaller systems to solve, without using any matrix inverse.]

Suppose Ais an \(n \times n\) matrix with the property that the equation \(Ax = 0\)has only the trivial solution. Without using the Invertible Matrix Theorem, explain directly why the equation \(Ax = b\) must have a solution for each b in \({\mathbb{R}^n}\).
See all solutions

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