Patternmaking for Fashion Design 4th Edition Pdf

Linear Algebra and Its Applications, Sixth Edition

By David C Lay, Steven R Lay and Judi J McDonald

Contents:

About the Authors 3

Preface 12

A Note to Students 22

Chapter i Linear Equations in Linear Algebra 25

INTRODUCTORY EXAMPLE: Linear Models in Economics

and Engineering 25

1.i Systems of Linear Equations 26

i.2 Row Reduction and Echelon Forms 37

1.3 Vector Equations 50

ane.4 The Matrix Equation Ax D b 61

one.v Solution Sets of Linear Systems 69

1.6 Applications of Linear Systems 77

1.7 Linear Independence 84

1.eight Introduction to Linear Transformations 91

1.nine The Matrix of a Linear Transformation 99

one.10 Linear Models in Concern, Science, and Engineering science 109

Projects 117

Supplementary Exercises 117

Chapter 2 Matrix Algebra 121

INTRODUCTORY EXAMPLE: Computer Models in Aircraft Design 121

two.1 Matrix Operations 122

ii.2 The Inverse of a Matrix 135

ii.3 Characterizations of Invertible Matrices 145

2.iv Partitioned Matrices 150

2.5 Matrix Factorizations 156

ii.6 The Leontief Input–Output Model 165

2.7 Applications to Computer Graphics 171

ii.8 Subspaces of Rn 179

2.ix Dimension and Rank 186

Projects 193

Supplementary Exercises 193

Chapter three Determinants 195

INTRODUCTORY Instance: Weighing Diamonds 195

iii.one Introduction to Determinants 196

3.2 Properties of Determinants 203

3.3 Cramer's Rule, Volume, and Linear Transformations 212

Projects 221

Supplementary Exercises 221

Chapter 4 Vector Spaces 225

INTRODUCTORY Case: Discrete-Time Signals and Digital

Signal Processing 225

four.1 Vector Spaces and Subspaces 226

4.2 Nix Spaces, Column Spaces, Row Spaces, and Linear

Transformations 235

4.three Linearly Contained Sets; Bases 246

4.4 Coordinate Systems 255

4.5 The Dimension of a Vector Space 265

4.six Change of Footing 273

4.7 Digital Point Processing 279

iv.8 Applications to Departure Equations 286

Projects 295

Supplementary Exercises 295

Chapter five Eigenvalues and Eigenvectors 297

INTRODUCTORY Example: Dynamical Systems and Spotted Owls 297

five.1 Eigenvectors and Eigenvalues 298

5.2 The Characteristic Equation 306

five.3 Diagonalization 314

five.4 Eigenvectors and Linear Transformations 321

v.5 Complex Eigenvalues 328

v.6 Discrete Dynamical Systems 335

5.7 Applications to Differential Equations 345

5.8 Iterative Estimates for Eigenvalues 353

v.9 Applications to Markov Chains 359

Projects 369

Supplementary Exercises 369

Chapter six Orthogonality and Least Squares 373

INTRODUCTORY EXAMPLE: Artificial Intelligence and Car

Learning 373

six.1 Inner Product, Length, and Orthogonality 374

6.two Orthogonal Sets 382

6.3 Orthogonal Projections 391

half-dozen.4 The Gram–Schmidt Procedure 400

vi.5 Least-Squares Problems 406

half dozen.6 Machine Learning and Linear Models 414

6.7 Inner Product Spaces 423

half dozen.8 Applications of Inner Product Spaces 431

Projects 437

Supplementary Exercises 438

Affiliate seven Symmetric Matrices and Quadratic Forms 441

INTRODUCTORY Case: Multichannel Image Processing 441

seven.1 Diagonalization of Symmetric Matrices 443

vii.ii Quadratic Forms 449

vii.3 Constrained Optimization 456

seven.4 The Singular Value Decomposition 463

7.v Applications to Image Processing and Statistics 473

Projects 481

Supplementary Exercises 481

Chapter 8 The Geometry of Vector Spaces 483

INTRODUCTORY EXAMPLE: The Platonic Solids 483

8.one Affine Combinations 484

8.2 Affine Independence 493

8.3 Convex Combinations 503

viii.4 Hyperplanes 510

8.five Polytopes 519

eight.half dozen Curves and Surfaces 531

Project 542

Supplementary Exercises 543

Chapter nine Optimization 545

INTRODUCTORY EXAMPLE: The Berlin Airlift 545

9.1 Matrix Games 546

ix.2 Linear Programming Geometric Method 560

nine.iii Linear Programming Simplex Method 570

9.4 Duality 585

Project 594

Supplementary Exercises 594

Chapter 10 Finite-State Markov Chains C-1

(Available Online)

INTRODUCTORY EXAMPLE: Googling Markov Chains C-1

10.1 Introduction and Examples C-2

10.2 The Steady-State Vector and Google's PageRank C-xiii

x.three Communication Classes C-25

10.4 Classification of States and Periodicity C-33

10.5 The Fundamental Matrix C-42

x.6 Markov Chains and Baseball Statistics C-54

Appendixes

A Uniqueness of the Reduced Echelon Grade 597

B Complex Numbers 599

Credits 604

Glossary 605

Answers to Odd-Numbered Exercises A-1

Index I-1

Preface

The response of students and teachers to the beginning v editions of Linear Algebra and Its Applications has been nearly gratifying. This Sixth Edition provides substantial support both for teaching and for using technology in the course. Equally before, the text provides a modern elementary introduction to linear algebra and a broad selection of interesting classical and leading-border applications. The material is accessible to students with the maturity that should come from successful completion of two semesters of college-level mathematics, unremarkably calculus.

The principal goal of the text is to assistance students main the bones concepts and skills they will utilize later in their careers. The topics here follow the recommendations of the original Linear Algebra Curriculum Study Group (LACSG), which were based on a careful investigation of the real needs of the students and a consensus amongst professionals in many disciplines that employ linear algebra. Ideas beingness discussed by the second Linear Algebra Curriculum Report Grouping (LACSG 2.0) have also been included. We hope this class volition be ane of the most useful and interesting mathematics classes taken by undergraduates.

What's New in This Edition

The 6th Edition has exciting new fabric, examples, and online resource. Afterward talking with high-tech industry researchers and colleagues in applied areas, we added new topics, vignettes, and applications with the intention of highlighting for students and faculty the linear algebraic foundational material for auto learning, bogus intelligence, data scientific discipline, and digital signal processing.

Content Changes

• Since matrix multiplication is a highly useful skill, we added new examples in Chapter 2 to show how matrix multiplication is used to identify patterns and scrub data. Respective exercises have been created to allow students to explore using matrix multiplication in various means.

• In our conversations with colleagues in manufacture and electrical engineering, we heard repeatedly how important understanding abstruse vector spaces is to their piece of work. Afterwards reading the reviewers' comments for Chapter iv, we reorganized the chapter, condensing some of the material on column, row, and nix spaces; moving Markov chains to the finish of Chapter 5; and creating a new department on signal processing. We view signals 12 every bit an infinite dimensional vector infinite and illustrate the usefulness of linear transformations to filter out unwanted "vectors" (a.k.a. dissonance), analyze information, and raise signals.

• By moving Markov chains to the end of Affiliate 5, we tin now discuss the steady land vector as an eigenvector. We also reorganized some of the summary material on determinants and change of basis to exist more specific to the way they are used in this chapter.

• In Chapter half dozen, we nowadays pattern recognition every bit an application of orthogonality, and the section on linear models now illustrates how machine learning relates to curve fitting.

• Chapter 9 on optimization was previously bachelor only as an online file. It has at present been moved into the regular textbook where information technology is more than readily bachelor to kinesthesia and students. Afterward an opening department on finding optimal strategies to two-person zerosum games, the rest of the chapter presents an introduction to linear programming— from two-dimensional issues that can be solved geometrically to higher dimensional bug that are solved using the Simplex Method.

Other Changes

• In the high-tech manufacture, where well-nigh computations are done on computers, judging the validity of data and computations is an important stride in preparing and analyzing information. In this edition, students are encouraged to learn to analyze their own computations to run across if they are consistent with the data at mitt and the questions being asked. For this reason, we take added "Reasonable Answers" advice and exercises to guide students.

• Nosotros accept added a list of projects to the terminate of each chapter (available online and in MyLab Math). Some of these projects were previously available online and take a wide range of themes from using linear transformations to create art to exploring additional ideas in mathematics. They can be used for grouping work or to raise the learning of individual students.

• PowerPoint lecture slides have been updated to encompass all sections of the text and cover them more thoroughly.

Distinctive Features Early Introduction of Key Concepts

Many fundamental ideas of linear algebra are introduced within the first 7 lectures, in the concrete setting of Rn, and then gradually examined from unlike points of view. Later generalizations of these concepts appear every bit natural extensions of familiar ideas, visualized through the geometric intuition developed in Chapter 1. A major achievement of this text is that the level of difficulty is fairly even throughout the course.

A Modern View of Matrix Multiplication

Good notation is crucial, and the text reflects the mode scientists and engineers actually use linear algebra in practice. The definitions and proofs focus on the columns of a matrix rather than on the matrix entries. A key theme is to view a matrix–vector product Ax every bit a linear combination of the columns of A. This mod approach simplifies many arguments, and it ties vector space ideas into the study of linear systems.

Linear Transformations

Linear transformations form a "thread" that is woven into the fabric of the text. Their use enhances the geometric flavor of the text. In Chapter 1, for instance, linear transformations provide a dynamic and graphical view of matrix–vector multiplication.

Eigenvalues and Dynamical Systems

Eigenvalues appear fairly early in the text, in Chapters v and 7. Because this material is spread over several weeks, students have more time than usual to absorb and review these disquisitional concepts. Eigenvalues are motivated by and applied to discrete and continuous dynamical systems, which appear in Sections 1.10, iv.8, and five.9, and in v sections of Chapter five. Some courses reach Chapter 5 after about five weeks past covering Sections two.8 and 2.9 instead of Chapter four. These two optional sections present all the vector space concepts from Chapter 4 needed for Chapter 5.

Orthogonality and To the lowest degree-Squares Problems

These topics receive a more comprehensive handling than is commonly found in offset texts. The original Linear Algebra Curriculum Study Group has emphasized the need for a substantial unit on orthogonality and least-squares problems, because orthogonality plays such an important function in calculator calculations and numerical linear algebra and considering inconsistent linear systems ascend so frequently in practical work.

Pedagogical Features Applications

A wide option of applications illustrates the power of linear algebra to explicate fundamental principles and simplify calculations in engineering, reckoner science, mathematics, physics, biology, economics, and statistics. Some applications appear in separate sections; others are treated in examples and exercises. In addition, each chapter opens with an introductory vignette that sets the stage for some application of linear algebra and provides a motivation for developing the mathematics that follows.

A Strong Geometric Emphasis

Every major concept in the course is given a geometric interpretation, because many students learn better when they tin can visualize an idea. There are substantially more than drawings here than usual, and some of the figures have never earlier appeared in a linear algebra text. Interactive versions of many of these figures announced in MyLab Math.

Examples

This text devotes a larger proportion of its expository material to examples than exercise most linear algebra texts. There are more examples than an instructor would ordinarily nowadays in class. But because the examples are written carefully, with lots of detail, students can read them on their own.

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