/*! 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} Q2E In Exercise 1-10, assume that聽\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Q2E (page 1)

In Exercise 1-10, assume that\(T\)is a linear transformation. Find the standard matrix of\(T\).

\(T:{\mathbb{R}^3} \to {\mathbb{R}^2}\), \(T\left( {{e_1}} \right) = \left( {1,3} \right)\) and \(T\left( {{e_2}} \right) = \left( {4, - 7} \right) - \) and \(T\left( {{e_3}} \right) = \left( { - 5,4} \right)\)where \({e_1}\), \({e_2}\), and \({e_3}\).

Short Answer

Expert verified

\(\left[ {\begin{array}{*{20}{c}}1&4&{ - 5}\\3&{ - 7}&4\end{array}} \right]\)

Step by step solution

01

Find the value of \(T\) using linear transformation

Using linear transformation,

\(\begin{aligned}{c}T &= T\left( {{x_1}{e_1} + {x_2}{e_2} + {x_3}{e_3}} \right)\\ &= {x_1}T\left( {{e_1}} \right) + {x_2}T\left( {{e_2}} \right) + {x_3}T\left( {{e_3}} \right)\\ &= \left[ {\begin{array}{*{20}{c}}{T\left( {{e_1}} \right)}&{T\left( {{e_2}} \right)}&{T\left( {{e_3}} \right)}\end{array}} \right]x\end{aligned}\)

02

Substitute values of \(T\left( {{e_1}} \right)\), \(T\left( {{e_2}} \right)\), and \(T\left( {{e_3}} \right)\)

By the equation \(T = \left[ {\begin{array}{*{20}{c}}{T\left( {{e_1}} \right)}&{T\left( {{e_2}} \right)}&{T\left( {{e_3}} \right)}\end{array}} \right]x\),

\(T = \left[ {\begin{array}{*{20}{c}}1&4&{ - 5}\\3&{ - 7}&4\end{array}} \right]x\).

03

Find the standard matrix \(T\) for linear transformation

In the equation \(T = Ax\), the matrix \(A\) is the matrix for linear transformation \(T\).

By the equation \(T = \left[ {\begin{array}{*{20}{c}}1&4&{ - 5}\\3&{ - 7}&4\end{array}} \right]x\), \(A = \left[ {\begin{array}{*{20}{c}}1&4&{ - 5}\\3&{ - 7}&4\end{array}} \right]\).

So, the linear transformation matrix is \(\left[ {\begin{array}{*{20}{c}}1&4&{ - 5}\\3&{ - 7}&4\end{array}} \right]\).

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

If Ais 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.

Suppose Tand Ssatisfy the invertibility equations (1) and (2), where T is a linear transformation. Show directly that Sis a linear transformation. (Hint: Given u, v in \({\mathbb{R}^n}\), let \({\mathop{\rm x}\nolimits} = S\left( {\mathop{\rm u}\nolimits} \right),{\mathop{\rm y}\nolimits} = S\left( {\mathop{\rm v}\nolimits} \right)\). Then \(T\left( {\mathop{\rm x}\nolimits} \right) = {\mathop{\rm u}\nolimits} \), \(T\left( {\mathop{\rm y}\nolimits} \right) = {\mathop{\rm v}\nolimits} \). Why? Apply Sto both sides of the equation \(T\left( {\mathop{\rm x}\nolimits} \right) + T\left( {\mathop{\rm y}\nolimits} \right) = T\left( {{\mathop{\rm x}\nolimits} + y} \right)\). Also, consider \(T\left( {cx} \right) = cT\left( x \right)\).)

Describe the possible echelon forms of the matrix A. Use the notation of Example 1 in Section 1.2.

a. A is a \({\bf{2}} \times {\bf{3}}\) matrix whose columns span \({\mathbb{R}^{\bf{2}}}\).

b. A is a \({\bf{3}} \times {\bf{3}}\) matrix whose columns span \({\mathbb{R}^{\bf{3}}}\).

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.

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

14. \(\left[ {\begin{array}{*{20}{c}}1&2&{ - 5}&{ - 6}&0&{ - 5}\\0&1&{ - 6}&{ - 3}&0&2\\0&0&0&0&1&0\\0&0&0&0&0&0\end{array}} \right]\).
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Calculus

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.