Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 1: Q19E (page 1)

In Exercises 17-20, show that \(T\) is a linear transformation by finding a matrix that implements the mapping. Note that \({x_1}\), \({x_2}\),……… are not vectors but are enteries in vectors

\(T\left( {{x_1},{x_2},{x_3}} \right) = \left( {{x_1} - 5{x_2} + 4{x_3},{x_2} - 6{x_3}} \right)\)

Short Answer

Expert verified

\(\left[ {\begin{array}{*{20}{c}}1&{ - 5}&4\\0&1&{ - 6}\end{array}} \right]\)

Step by step solution

01

Express \(T\left( x \right)\) in the form of a matrix

Write the linear transformation \(T\left( x \right)\).

\(T\left( x \right) = \left[ {\begin{array}{*{20}{c}}{{x_1} - 5{x_2} + 4{x_3}}\\{{x_2} - 6{x_3}}\end{array}} \right]\)

02

Solve the equation \(T\left( x \right) = Ax\)

\(\left[ {\begin{array}{*{20}{c}}{{x_1} - 5{x_2} + 4{x_3}}\\{{x_2} - 6{x_3}}\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right]\)

As \(T\left( x \right)\) has only two entries, matrix \(A\) will have two rows and \(x\) has three entries so matrix A will have three columns.

03

Compare the rows of the matrix

From the equation \(\left[ {\begin{array}{*{20}{c}}{{x_1} - 5{x_2} + 4{x_3}}\\{{x_2} - 6{x_3}}\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right]\), the first row of matrix \(A\) is \(\left[ {\begin{array}{*{20}{c}}1&{ - 5}&4\end{array}} \right]\).

04

Compare the rows of the matrix

From the equation \(\left[ {\begin{array}{*{20}{c}}{{x_1} - 5{x_2} + 4{x_3}}\\{{x_2} - 6{x_3}}\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right]\), the second row of matrix \(A\) is \(\left[ {\begin{array}{*{20}{c}}0&1&{ - 6}\end{array}} \right]\).

So, the matrix given in the equation is \(\left[ {\begin{array}{*{20}{c}}1&{ - 5}&4\\0&1&{ - 6}\end{array}} \right]\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the general solutions of the systems whose augmented matrices are given

11. \(\left[ {\begin{array}{*{20}{c}}3&{ - 4}&2&0\\{ - 9}&{12}&{ - 6}&0\\{ - 6}&8&{ - 4}&0\end{array}} \right]\).

In Exercises 5, write a system of equations that is equivalent to the given vector equation.

5. \({x_1}\left[ {\begin{array}{*{20}{c}}6\\{ - 1}\\5\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}{ - 3}\\4\\0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 7}\\{ - 5}\end{array}} \right]\)

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.

In Exercises 23 and 24, key statements from this section are either quoted directly, restated slightly (but still true), or altered in some way that makes them false in some cases. Mark each statement True or False, and justify your answer.(If true, give the approximate location where a similar statement appears, or refer to a definition or theorem. If false, give the location of a statement that has been quoted or used incorrectly, or cite an example that shows the statement is not true in all cases.) Similar true/false questions will appear in many sections of the text.

24.

a. Elementary row operations on an augmented matrix never change the solution set of the associated linear system.

b. Two matrices are row equivalent if they have the same number of rows.

c. An inconsistent system has more than one solution.

d. Two linear systems are equivalent if they have the same solution set.

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

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free