Solutions Manual for Linear Algebra and Its Applications 6th Edition By David Lay, Steven Lay, Judi

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INSTRUCTOR’S SOLUTIONS MANUAL LINEAR ALGEBRA AND ITS APPLICATIONS SIXTH EDITION

David C. Lay University of Maryland–College Park

Steven R. Lay Lee University

Judi J. McDonald Washington State University

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Contents Introduction

v

Chapter 1 Linear Equations in Linear Algebra 1.1 Systems of Linear Equations 1.2 Row Reduction and Echelon Forms 1.3 Vector Equations 1.4 The Matrix Equation Ax = b 1.5 Solution Sets of Linear Systems 1.6 Applications of Linear Systems 1.7 Linear Independence 1.8 Introduction to Linear Transformations 1.9 The Matrix of a Linear Transformation 1.10 Linear Models in Business, Science, and Engineering Supplementary Exercises

Chapter 2 Matrix Algebra

1-1 1-1 1-8 1-16 1-25 1-33 1-42 1-51 1-58 1-65 1-71 1-80

2-1

2.1 Matrix Operations 2.2 The Inverse of a Matrix 2.3 Characterization of Invertible Matrices 2.4 Partitioned Matrices 2.5 Matrix Factorizations 2.6 The Leontief Input-Output Model 2.7 Applications to Computer Graphics 2.8 Subspaces of n 2.9 Dimension and Rank Supplementary Exercises

Chapter 3 Determinants

2-1 2-7 2-15 2-23 2-32 2-47 2-51 2-58 2-66 2-72

3-1

3.1 Introduction to Determinants 3.2 Properties of Determinants 3.3 Cramer’s Rule, Volume, and Linear Transformations Supplementary Exercises

Chapter 4 Vector Spaces

3-1 3-8 3-14 3-22

4-1

4.1 Vector Spaces and Subspaces 4-1 4.2 Null Spaces, Column Spaces, Row Spaces, and Linear Transformations 4-7 4.3 Linearly Independent Sets; Bases 4-15 4.4 Coordinate Systems 4-23 4.5 The Dimension of a Vector Space 4-30 4.6 Change of Basis 4-36 4.7 Digital Signal Processing 4-40 4.8 Applications to Difference Equations 4-43 Supplementary Exercises 4-52

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Chapter 5 Eigenvalues and Eigenvectors 5.1 Eigenvalues and Eigenvectors 5.2 The Characteristic Equation 5.3 Diagonalization 5.4 Eigenvalues and Linear Transformations 5.5 Complex Eigenvalues 5.6 Discrete Dynamical Systems 5.7 Applications to Differential Equations 5.8 Iterative Estimates for Eigenvalues 5.9 Applications to Markov Chains Supplementary Exercises

Chapter 6 Orthogonality and Least Squares 6.1 Inner Product, Length, and Orthogonality 6.2 Orthogonal Sets 6.3 Orthogonal Projections 6.4 The Gram-Schmidt Process 6.5 Least-Squares Problems 6.6 Machine Learning and Linear Models 6.7 Inner Product Spaces 6.8 Applications of Inner Product Spaces Supplementary Exercises

Chapter 7 Symmetric Matrices and Quadratic Forms 7.1 Diagonalization of Symmetric Matrices 7.2 Quadratic Forms 7.3 Constrained Optimization 7.4 The Singular Value Decomposition 7.5 Applications to Image Processing and Statistics Supplementary Exercises

5-1 5-1 5-10 5-15 5-29 5-35 5-43 5-49 5-59 5-67 5-75

6-1 6-1 6-5 6-10 6-18 6-24 6-29 6-34 6-38 6-43

7-1 7-1 7-14 7-22 7-27 7-37 7-40

Chapter 8 The Geometry of Vector Spaces

8-1

8.1 Affine Combinations 8.2 Affine Independence 8.3 Convex Combinations 8.4 Hyperplanes 8.5 Polytopes 8.6 Curves and Surfaces Supplementary Exercises

8-1 8-5 8-10 8-15 8-19 8-22 8-27

Chapter 9 Optimization

9-1

9.1 Matrix Games 9.2 Linear Programming—Geometric Method 9.3 Linear Programming—Simplex Method 9.4 Duality Supplementary Exercises

9-1 9-7 9-11 9-15 9-22

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Introduction I fell in love with linear algebra when I was an undergraduate student and it has remained a central part of my life since that time. It is an interesting and beautiful subject, with a broad range of applications. In recent years, I consistently hear from industry partners about how much the high-tech industry appreciates individuals having a strong foundation in both technical and theoretical aspects of linear algebra. I hope you will enjoy teaching this course as much as I do. You are also welcome to email me at LLinearAlgebra@gmail.com any time you have comments and suggestions, or just want to talk about linear algebra.

There are many ways in which modern technology can support (or hinder) your student’s learning. The interactive figures from the electronic textbook can be used in classroom demonstrations to bring linear algebraic concepts alive and demonstrate numerous examples with the push of a button. Take time to explore with the interactive figures and show your students how to use technology to find key definitions and theorems quickly in the electronic textbook.

If your course uses MyLab for homework, there are several things to be aware of.

First, for some exercises, your students will enter only a final answer. To get to that answer, they may have half a page or more of calculations. Encourage them to keep a notebook with the exercise statement, worked solutions, and summary notes about what they learned while solving an exercise. As an instructor, you can choose many settings in the program. You can set how many tries students are allowed to solve each question. Most exercises let the student have three tries before MyLab either records an incorrect answer or offers the student a similar question. In my experience, persistence pays off – if students are allowed to continue to work similar exercises, mastery of the skill will result. I have also found that the “View an Example” and “Help Me Solve It” tab help get students going again when they are stuck.

At the end of each chapter, we have highlighted some of the projects that are available online, but moved away from updating the toolbox and the computer manuals. I find that when I am trying to code almost anything, I go to the help features for the program or open a search engine and enter some key words. There are still tips in the Student Study Guide about appropriate MATLAB code for various parts of the course.

Technology also provides students with easy access to a wealth of videos on linear algebra and solutions for some of the exercises. Please refrain from posting portions of this Instructor’s Solution Manual online, as by doing so you are giving other instructors’ students solutions to the exercises. Some of the open-access online videos are amazing. Others contain errors or introduce the material in a different order from how it is covered in this text, leading to confusion. I try to talk to my students about using technology to learn effectively without it becoming a crutch that leaves them with perfect homework and failed exams.

The Instructor’s Solution Manual contains detailed solutions for all the exercises, as well as advice on the exercises themselves. I am interested to hear from you at LLinearAlgebra@gmail.com as to what types of material you would like to use in your course, additional topics you would like to see covered, any typos you find, or just to talk about my favorite subject – linear algebra.

—Judi J. McDonald

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1.1 - Systems Of Linear Equations Notes: The key exercises are 7 (or 11 or 12), 23–26, and 35. For brevity, the symbols R1, R2,…, stand for row 1 (or equation 1), row 2 (or equation 2), and so on. Additional notes are at the end of the section. In Exercises 15–18, students are asked to check their answers to Exercises 11–14; checking that solutions are correct, or at least reasonable, is an important skill in the high-tech industry.

1.

x1 + 5x2 = 7 −2x1 − 7 x2 = −5

 1  −2 

5 −7

7 −5

Replace R2 by R2 + (2)R1 and obtain:

x1 + 5x2 = 7 3x2 = 9 x1 + 5x2 = 7

Scale R2 by 1/3:

x2 = 3 x1

Replace R1 by R1 + (–5)R2:

= −8 x2 = 3

1 0 

5 3

7 9

1 0 

5 1

7 3

 1 0 −8 0 1 3 

The solution is (x1, x2) = (–8, 3), or simply (–8, 3).

2.

2 x1 + 4 x2 = −4 5x1 + 7 x2 = 11

2 5 

4 7

Scale R1 by 1/2 and obtain: Replace R2 by R2 + (–5)R1: Scale R2 by –1/3: Replace R1 by R1 + (–2)R2:

−4 11 

x1 + 2 x2 = −2 5x1 + 7 x2 = 11 x1 + 2x2 = −2 −3x2 = 21 x1 + 2 x2 = −2 x2 = −7 x1

= 12 x2 = −7

1 5 

2 7

−2 11 

1 0 

2 −3

−2 21

 1 2 −2 0 1 −7     1 0 12 0 1 −7  

The solution is (x1, x2) = (12, –7), or simply (12, –7).

3. The point of intersection satisfies the system of two linear equations: 1-1 Copyright © 2021 Pearson Education, Inc.

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1-2

Chapter 1



Linear Equations in Linear Algebra

x1 + 5x2 = 7 x1 − 2x2 = −2

1 1 

5 −2

7 −2

Replace R2 by R2 + (–1)R1 and obtain:

x1 + 5x2 = 7 −7 x2 = −9 x1 + 5x2 = 7

Scale R2 by –1/7:

x2 = 9/7 x1

Replace R1 by R1 + (–5)R2:

= 4/7 x2 = 9/7

1 0 

5 −7

7 −9

1 0 

5 1

7  9/7 

1 0 

0 1

4/7  9/7 

 1 −5 0 8 

1 2

The point of intersection is (x1, x2) = (4/7, 9/7).

4. The point of intersection satisfies the system of two linear equations:

x1 − 5x2 = 1 3x1 − 7 x2 = 5

1 3 

−5 −7

1 5

x1 − 5x2 = 1

Replace R2 by R2 + (–3)R1 and obtain:

8x2 = 2 x1 − 5x2 = 1

Scale R2 by 1/8:

x2 = 1/4 x1

Replace R1 by R1 + (5)R2:

= 9/4 x2 = 1/4

1  1 −5 0 1 1/4  1 0 

0 1

9/4  1/4 

The point of intersection is (x1, x2) = (9/4, 1/4).

5. The system is already in “triangular” form. The fourth equation is x4 = –5, and the other equations do not contain the variable x4. The next two steps should be to use the variable x3 in the third equation to eliminate that variable from the first two equations. In matrix notation, that means to replace R2 by its sum with 3 times R3, and then replace R1 by its sum with –5 times R3.

6. One more step will put the system in triangular form. Replace R4 by its sum with –3 times R3, which 4 0 −1  1 −6 0 2 −7 0 4  . After that, the next step is to scale the fourth row by –1/5. produces  0 0 1 2 −3   0 0 −5 15 0 7. Ordinarily, the next step would be to interchange R3 and R4, to put a 1 in the third row and third column. But in this case, the third row of the augmented matrix corresponds to the equation 0 x1 + 0 x2 + 0 x3 = 1 , or simply, 0 = 1. A system containing this condition has no solution. Further row operations are unnecessary once an equation such as 0 = 1 is evident.

The solution set is empty.

8. The standard row operations begin by multiplying R3 by ½. Next, replace R2 by R2+ (–7)R3 and R1

by R1+ (–2)R3. Finally, replace R1 by R1+ (–1)R2.

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