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Linear Algebra and its Applications
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Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

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

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

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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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: 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 \(A = \left( {\begin{aligned}{*{20}{c}}6&{ - 8}\\{ - 5}&7\end{aligned}} \right)\).

Step 2: Show that T  is invertible

Theorem 4 states 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 A is not invertible.

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

Step 3: Determine the formula for \({T^{ - 1}}\)

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be a linear transformation and A be the standard matrix for T. Then, according to Theorem 9, T is invertible if and only if A is 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 T is 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 \({T^{ - 1}}\left( {{x_1},{x_2}} \right) = \left( {\frac{7}{2}{x_1} + 4{x_2},\frac{5}{2}{x_1} + 3{x_2}} \right)\).

Most popular questions for Math Textbooks

In (a) and (b), suppose the vectors are linearly independent. What can you say about the numbers \(a,....,f\) ? Justify your answers. (Hint: Use a theorem for (b).)

  1. \(\left( {\begin{aligned}{*{20}{c}}a\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\d\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\end{aligned}} \right)\)
  2. \(\left( {\begin{aligned}{*{20}{c}}a\\1\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\1\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\\1\end{aligned}} \right)\)

If A is an \(n \times n\) matrix and the transformation \({\bf{x}}| \to A{\bf{x}}\) is one-to-one, what else can you say about this transformation? Justify your answer.

Write the vector \(\left( {\begin{array}{*{20}{c}}5\\6\end{array}} \right)\) as the sum of two vectors, one on the line \(\left\{ {\left( {x,y} \right):y = {\bf{2}}x} \right\}\) and one on the line \(\left\{ {\left( {x,y} \right):y = x/{\bf{2}}} \right\}\).

Consider the dynamical system x→(t+1)=[1.100⅄]X→(t).

Sketch a phase portrait of this system for the given values of λ :

λ=1

Consider the problem of determining whether the following system of equations is consistent:

\(\begin{aligned}{c}{\bf{4}}{x_1} - {\bf{2}}{x_2} + {\bf{7}}{x_3} = - {\bf{5}}\\{\bf{8}}{x_1} - {\bf{3}}{x_2} + {\bf{10}}{x_3} = - {\bf{3}}\end{aligned}\)

  1. Define appropriate vectors, and restate the problem in terms of linear combinations. Then solve that problem.

  1. Define an appropriate matrix, and restate the problem using the phrase “columns of A.”

  1. Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T.
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