/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q17E In Exercises 17-20, show that \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Q17E (page 1)

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

\(T\left( {{x_1},{x_2},{x_3},{x_4}} \right) = \left( {0,{x_1} + {x_2},{x_2} + {x_3},{x_3} + {x_4}} \right)\)

Short Answer

Expert verified

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

Step by step solution

01

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

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

\(T\left( x \right) = \left[ {\begin{array}{*{20}{c}}0\\{{x_1} + {x_2}}\\{{x_2} + {x_3}}\\{{x_3} + {x_4}}\end{array}} \right]\)

02

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

\(\left[ {\begin{array}{*{20}{c}}0\\{{x_1} + {x_2}}\\{{x_2} + {x_3}}\\{{x_3} + {x_4}}\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\\{{x_4}}\end{array}} \right]\)

By matrix multiplication, the order of matrix \(A\) is \(4 \times 4\).

03

Compare the rows of the matrix

From the equation \(\left[ {\begin{array}{*{20}{c}}0\\{{x_1} + {x_2}}\\{{x_2} + {x_3}}\\{{x_3} + {x_4}}\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\\{{x_4}}\end{array}} \right]\), the first row of matrix \(A\) is \(\left[ {\begin{array}{*{20}{c}}0&0&0&0\end{array}} \right]\).

04

Compare the rows of the matrix

From the equation \(\left( {\begin{array}{*{20}{c}}0\\{{x_1} + {x_2}}\\{{x_2} + {x_3}}\\{{x_3} + {x_4}}\end{array}} \right) = \left( A \right)\left( {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\\{{x_4}}\end{array}} \right)\), the third row of matrix \(A\) is \(\left( {\begin{array}{*{20}{c}}0&1&1&0\end{array}} \right)\).

05

Compare the rows of the matrix

From the equation \(\left[ {\begin{array}{*{20}{c}}0\\{{x_1} + {x_2}}\\{{x_2} + {x_3}}\\{{x_3} + {x_4}}\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\\{{x_4}}\end{array}} \right]\), the third row of matrix \(A\) is \(\left[ {\begin{array}{*{20}{c}}0&1&1&0\end{array}} \right]\).

06

Compare the rows of the matrix

From the equation \(\left[ {\begin{array}{*{20}{c}}0\\{{x_1} + {x_2}}\\{{x_2} + {x_3}}\\{{x_3} + {x_4}}\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\\{{x_4}}\end{array}} \right]\), the third row of matrix \(A\) is \(\left[ {\begin{array}{*{20}{c}}0&0&1&1\end{array}} \right]\).

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

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

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

In Exercises 13 and 14, determine if \(b\) is a linear combination of the vectors formed from the columns of the matrix \(A\).

13. \(A = \left[ {\begin{array}{*{20}{c}}1&{ - 4}&2\\0&3&5\\{ - 2}&8&{ - 4}\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}3\\{ - 7}\\{ - 3}\end{array}} \right]\)

In Exercises 31, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

31. \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\4&{ - 1}&3&{ - 6}\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\0&7&{ - 1}&{ - 6}\end{array}} \right]\)

In Exercises 15 and 16, list five vectors in Span \(\left\{ {{v_1},{v_2}} \right\}\). For each vector, show the weights on \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) used to generate the vector and list the three entries of the vector. Do not make a sketch.

16. \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}3\\0\\2\end{array}} \right],{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right]\)

Determine the values(s) of \(h\) such that matrix is the augmented matrix of a consistent linear system.

18. \(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&{ - 2}\\5&h&{ - 7}\end{array}} \right]\)

In Exercise 23 and 24, make each statement True or False. Justify each answer.

23.

a. Another notation for the vector \(\left[ {\begin{array}{*{20}{c}}{ - 4}\\3\end{array}} \right]\) is \(\left[ {\begin{array}{*{20}{c}}{ - 4}&3\end{array}} \right]\).

b. The points in the plane corresponding to \(\left[ {\begin{array}{*{20}{c}}{ - 2}\\5\end{array}} \right]\) and \(\left[ {\begin{array}{*{20}{c}}{ - 5}\\2\end{array}} \right]\) lie on a line through the origin.

c. An example of a linear combination of vectors \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) is the vector \(\frac{1}{2}{{\mathop{\rm v}\nolimits} _1}\).

d. The solution set of the linear system whose augmented matrix is \(\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}&b\end{array}} \right]\) is the same as the solution set of the equation\({{\mathop{\rm x}\nolimits} _1}{a_1} + {x_2}{a_2} + {x_3}{a_3} = b\).

e. The set Span \(\left\{ {u,v} \right\}\) is always visualized as a plane through the origin.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Pure 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.