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by David C. Lay

Cover type: HardbackEdition: 3RD 03

Copyright: 2003

Publisher: Addison-Wesley Longman, Inc.

Published: 2003

International: No

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Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations), are not easily understood, and require time to assimilate. Since they are fundamental to the study of linear algebra, students' understanding of these concepts is vital to their mastery of the subject. Lay introduces these concepts early in a familiar, concrete Rn setting, develops them gradually, and returns to them again and again throughout the text. Finally, when discussed in the abstract, these concepts are more accessible.

**New To This Edition :**

- New full color design allows for clearer understanding of figures and graphically important concepts and procedures.
- Section 1.6, Applications of Linear Systems, is new to this edition. It includes applications on the Leontief Economic Model, Solving Chemical Equations, and Traffic Flow.
- Section 2.9 from the Updated 2/e, Subspaces of R n, has been split into two sections, 2.8 and 2.9. This forms a better bridge to Chapter 5 for those instructors who wish to cover Eigenvalues earlier in the course, or postpone or omit vector spaces. This option is ideal for a shorter course.
- A new Instructor's Solutions Manual contains detailed solutions for all exercises, including problem-specific teaching notes.
- MyMathLab is now available, integrating the text's content with the Student Study Guide. All of the text's many electronic resources can be found on MyMathLab.
- MathXL is now available for the text, allowing students to take tests and quizzes online.
- An electronic test generator, TestGen-EQ, allows for creation of paper tests, or can be used in conjunction with MathXL to test or do homework online.

**Features :**

- MyMathLab
- Interactive Content. The full text is now available online, integrating the text's content with the Student's Study Guide and electronic practice problems. All of the text's many electronic resources can be found on MyMathLab.
- MathXL allows students to take tests and quizzes online. It works in conjunction with TestGen-EQ, allowing instructors to post custom tests for quizzing, homework, or practice.
- Student's Study Guide is now available free as a part of MyMathLab.
- Transparency Masters for every chapter help save time. For instructors only.
- Downloadable Data Files for exercises and projects are available for Matlab, Maple, Mathematica, and selected graphing calculators.
- Sample Syllabi written by David C. Lay offer suggestions for course organization. For instructors only.
- Review Sheets and Practice Tests help students study effectively.
- Sample Tests with solutions written by David C. Lay provide classroom-tested exam options for instructors. For instructors only.
- Applications by Chapter are a selection of case studies, problems, and application projects based on real-world data and keyed to the text.
- References to Applications lists hundreds of applications referred to in the text, and provides hot links to the case studies and application projects.
- Teaching Notes for Linear Algebra: Modules for Interactive Learning Using Maple®, is an interactive linear algebra tutorial. These highly geometric modules can be used for classroom demonstrations, individual student work or interactive collaboration. For instructors only.
- Fundamental ideas of linear algebra are introduced within the first seven lectures, in the concrete setting of R n, and then gradually examined from different points of view. Later generalizations of these concepts appear as natural extensions of familiar ideas.
- Focus on visualization of concepts throughout the book.
- Icons in the margins to flag topics for which expanded or enhanced material is available on the Web.
- A modern view of matrix multiplication is presented. Definitions and proofs focus on the columns of a matrix rather than on the matrix entries.
- Numerical Notes give a realistic flavor to the text. Students are reminded frequently of issues that arise in the real-life use of linear algebra.
- Each major concept in the course is given a geometric interpretation because many students learn better when they can visualize an idea.
- M exercises appear in every section. To be solved with the aid of a Matrix program such as MATLAB, Maple, Mathematica, MathCad, Derive or programmable calculators with matrix capabilities, such as the TI-83 Plus, TI-86, TI-89, and HP-48G. Data for these exercises are provided on the Web.

**Lay, David C. : University of Maryland**

David C. Lay holds a B.A. from Aurora University (Illinois), and an M.A. and Ph.D. from the University of California at Los Angeles. Lay has been an educator and research mathematician since 1966, mostly at the University of Maryland, College Park. He has also served as a visiting professor at the University of Amsterdam, the Free University in Amsterdam, and the University of Kaiserslautern, Germany. He has over 30 research articles published in functional analysis and linear algebra.

As a founding member of the NSF-sponsored Linear Algebra Curriculum Study Group, Lay has been a leader in the current movement to modernize the linear algebra curriculum. Lay is also co-author of several mathematics texts, including Introduction to Functional Analysis, with Angus E. Taylor, Calculus and Its Applications, with L.J. Goldstein and D.I. Schneider, and Linear Algebra Gems-Assets for Undergraduate Mathematics, with D. Carlson, C.R. Johnson, and A.D. Porter.

A top-notch educator, Professor Lay has received four university awards for teaching excellence, including, in 1996, the title of Distinguished Scholar-Teacher of the University of Maryland. In 1994, he was given one of the Mathematical Association of America's Awards for Distinguished College or Unviersity Teaching of Mathematics. He has been elected by the university students to membership in Alpha Lambda Delta National Scholastic Honor Society and Golden Key National Honor Society. In 1989, Aurora University conferred on him the Outstanding Alumnus award. Lay is a member of the American Mathematical Society, the Canadian Mathematical Society, the International Linear Algebra Society, the Mathematical Association of America, Sigma Xi, and the Society for Industrial and Applied Mathematics. Since 1992, he has served several terms on the national board of the Association of Christians in the Mathematical Sciences.

*(Supplementary Exercises are featured at the end of each chapter.) *

1. Linear Equations in Linear Algebra.

Introductory Example: Linear Models in Economics and Engineering.

Systems of Linear Equations.

Row Reduction and Echelon Forms.

Vector Equations.

The Matrix Equation Ax = b.

Solution Sets of Linear Systems.

Applications of Linear Systems.

Linear Independence.

Introduction to Linear Transformations.

The Matrix of a Linear Transformation.

Linear Models in Business, Science, and Engineering.

2. Matrix Algebra.

Introductory Example: Computer Graphics in Aircraft Design.

Matrix Operations.

The Inverse of a Matrix.

Characterizations of Invertible Matrices.

Partitioned Matrices.

Matrix Factorizations.

The Leontief Input-Output Model.

Applications to Computer Graphics.

Subspaces of R n.

Dimensions and Rank.

3. Determinants.

Introductory Example: Determinants in Analytic Geometry.

Introduction to Determinainants.

Properties of Determinants.

Cramer's Rule, Volume, and Linear Transformations.

4. Vector Spaces.

Introductory Example: Space Flight and Control Systems.

Vector Spaces and Subspaces.

Null Spaces, Column Spaces, and Linear Transformations.

Linearly Independent Sets; Bases.

Coordinate Systems.

The Dimension of Vector Space

Rank.

Change of Basis.

Applications to Difference Equations.

Applications to Markov Chains.

5. Eigenvalues and Eigenvectors.

Introductory Example: Dynamical Systems and Spotted Owls.

Eigenvectors and Eigenvalues.

The Characteristic Equation.

Diagonalization.

Eigenvectors and Linear Transformations.

Complex Eigenvalues.

Discrete Dynamical Systems.

Applications to Differential Equations.

Iterative Estimates for Eigenvalues.

6. Orthogonality and Least-Squares.

Introductory Example: Readjusting the North American Datum.

Inner Product, Length, and Orthogonality.

Orthogonal Sets.

Orthogonal Projections.

The Gram-Schmidt Process.

Least-Squares Problems.

Applications to Linear Models.

Inner Product Spaces.

Applications of Inner Product Spaces.

7. Symmetric Matrices and Quadratic Forms.

Introductory Example: Multichannel Image Processing.

Diagonalization of Symmetric Matrices.

Quadratic Forms.

Constrained Optimization.

The Singular Value Decomposition.

Applications to Image Processing and Statistics.

Appendices.

A. Uniqueness of the Reduced Echelon Form.

B. Complex Numbers

Glossary.

Answers.

Index.

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Summary

Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations), are not easily understood, and require time to assimilate. Since they are fundamental to the study of linear algebra, students' understanding of these concepts is vital to their mastery of the subject. Lay introduces these concepts early in a familiar, concrete Rn setting, develops them gradually, and returns to them again and again throughout the text. Finally, when discussed in the abstract, these concepts are more accessible.

**New To This Edition :**

- New full color design allows for clearer understanding of figures and graphically important concepts and procedures.
- Section 1.6, Applications of Linear Systems, is new to this edition. It includes applications on the Leontief Economic Model, Solving Chemical Equations, and Traffic Flow.
- Section 2.9 from the Updated 2/e, Subspaces of R n, has been split into two sections, 2.8 and 2.9. This forms a better bridge to Chapter 5 for those instructors who wish to cover Eigenvalues earlier in the course, or postpone or omit vector spaces. This option is ideal for a shorter course.
- A new Instructor's Solutions Manual contains detailed solutions for all exercises, including problem-specific teaching notes.
- MyMathLab is now available, integrating the text's content with the Student Study Guide. All of the text's many electronic resources can be found on MyMathLab.
- MathXL is now available for the text, allowing students to take tests and quizzes online.
- An electronic test generator, TestGen-EQ, allows for creation of paper tests, or can be used in conjunction with MathXL to test or do homework online.

**Features :**

- MyMathLab
- Interactive Content. The full text is now available online, integrating the text's content with the Student's Study Guide and electronic practice problems. All of the text's many electronic resources can be found on MyMathLab.
- MathXL allows students to take tests and quizzes online. It works in conjunction with TestGen-EQ, allowing instructors to post custom tests for quizzing, homework, or practice.
- Student's Study Guide is now available free as a part of MyMathLab.
- Transparency Masters for every chapter help save time. For instructors only.
- Downloadable Data Files for exercises and projects are available for Matlab, Maple, Mathematica, and selected graphing calculators.
- Sample Syllabi written by David C. Lay offer suggestions for course organization. For instructors only.
- Review Sheets and Practice Tests help students study effectively.
- Sample Tests with solutions written by David C. Lay provide classroom-tested exam options for instructors. For instructors only.
- Applications by Chapter are a selection of case studies, problems, and application projects based on real-world data and keyed to the text.
- References to Applications lists hundreds of applications referred to in the text, and provides hot links to the case studies and application projects.
- Teaching Notes for Linear Algebra: Modules for Interactive Learning Using Maple®, is an interactive linear algebra tutorial. These highly geometric modules can be used for classroom demonstrations, individual student work or interactive collaboration. For instructors only.
- Fundamental ideas of linear algebra are introduced within the first seven lectures, in the concrete setting of R n, and then gradually examined from different points of view. Later generalizations of these concepts appear as natural extensions of familiar ideas.
- Focus on visualization of concepts throughout the book.
- Icons in the margins to flag topics for which expanded or enhanced material is available on the Web.
- A modern view of matrix multiplication is presented. Definitions and proofs focus on the columns of a matrix rather than on the matrix entries.
- Numerical Notes give a realistic flavor to the text. Students are reminded frequently of issues that arise in the real-life use of linear algebra.
- Each major concept in the course is given a geometric interpretation because many students learn better when they can visualize an idea.
- M exercises appear in every section. To be solved with the aid of a Matrix program such as MATLAB, Maple, Mathematica, MathCad, Derive or programmable calculators with matrix capabilities, such as the TI-83 Plus, TI-86, TI-89, and HP-48G. Data for these exercises are provided on the Web.

Author Bio

**Lay, David C. : University of Maryland**

David C. Lay holds a B.A. from Aurora University (Illinois), and an M.A. and Ph.D. from the University of California at Los Angeles. Lay has been an educator and research mathematician since 1966, mostly at the University of Maryland, College Park. He has also served as a visiting professor at the University of Amsterdam, the Free University in Amsterdam, and the University of Kaiserslautern, Germany. He has over 30 research articles published in functional analysis and linear algebra.

As a founding member of the NSF-sponsored Linear Algebra Curriculum Study Group, Lay has been a leader in the current movement to modernize the linear algebra curriculum. Lay is also co-author of several mathematics texts, including Introduction to Functional Analysis, with Angus E. Taylor, Calculus and Its Applications, with L.J. Goldstein and D.I. Schneider, and Linear Algebra Gems-Assets for Undergraduate Mathematics, with D. Carlson, C.R. Johnson, and A.D. Porter.

A top-notch educator, Professor Lay has received four university awards for teaching excellence, including, in 1996, the title of Distinguished Scholar-Teacher of the University of Maryland. In 1994, he was given one of the Mathematical Association of America's Awards for Distinguished College or Unviersity Teaching of Mathematics. He has been elected by the university students to membership in Alpha Lambda Delta National Scholastic Honor Society and Golden Key National Honor Society. In 1989, Aurora University conferred on him the Outstanding Alumnus award. Lay is a member of the American Mathematical Society, the Canadian Mathematical Society, the International Linear Algebra Society, the Mathematical Association of America, Sigma Xi, and the Society for Industrial and Applied Mathematics. Since 1992, he has served several terms on the national board of the Association of Christians in the Mathematical Sciences.

Table of Contents

*(Supplementary Exercises are featured at the end of each chapter.) *

1. Linear Equations in Linear Algebra.

Introductory Example: Linear Models in Economics and Engineering.

Systems of Linear Equations.

Row Reduction and Echelon Forms.

Vector Equations.

The Matrix Equation Ax = b.

Solution Sets of Linear Systems.

Applications of Linear Systems.

Linear Independence.

Introduction to Linear Transformations.

The Matrix of a Linear Transformation.

Linear Models in Business, Science, and Engineering.

2. Matrix Algebra.

Introductory Example: Computer Graphics in Aircraft Design.

Matrix Operations.

The Inverse of a Matrix.

Characterizations of Invertible Matrices.

Partitioned Matrices.

Matrix Factorizations.

The Leontief Input-Output Model.

Applications to Computer Graphics.

Subspaces of R n.

Dimensions and Rank.

3. Determinants.

Introductory Example: Determinants in Analytic Geometry.

Introduction to Determinainants.

Properties of Determinants.

Cramer's Rule, Volume, and Linear Transformations.

4. Vector Spaces.

Introductory Example: Space Flight and Control Systems.

Vector Spaces and Subspaces.

Null Spaces, Column Spaces, and Linear Transformations.

Linearly Independent Sets; Bases.

Coordinate Systems.

The Dimension of Vector Space

Rank.

Change of Basis.

Applications to Difference Equations.

Applications to Markov Chains.

5. Eigenvalues and Eigenvectors.

Introductory Example: Dynamical Systems and Spotted Owls.

Eigenvectors and Eigenvalues.

The Characteristic Equation.

Diagonalization.

Eigenvectors and Linear Transformations.

Complex Eigenvalues.

Discrete Dynamical Systems.

Applications to Differential Equations.

Iterative Estimates for Eigenvalues.

6. Orthogonality and Least-Squares.

Introductory Example: Readjusting the North American Datum.

Inner Product, Length, and Orthogonality.

Orthogonal Sets.

Orthogonal Projections.

The Gram-Schmidt Process.

Least-Squares Problems.

Applications to Linear Models.

Inner Product Spaces.

Applications of Inner Product Spaces.

7. Symmetric Matrices and Quadratic Forms.

Introductory Example: Multichannel Image Processing.

Diagonalization of Symmetric Matrices.

Quadratic Forms.

Constrained Optimization.

The Singular Value Decomposition.

Applications to Image Processing and Statistics.

Appendices.

A. Uniqueness of the Reduced Echelon Form.

B. Complex Numbers

Glossary.

Answers.

Index.

Publisher Info

Publisher: Addison-Wesley Longman, Inc.

Published: 2003

International: No

Published: 2003

International: No

Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations), are not easily understood, and require time to assimilate. Since they are fundamental to the study of linear algebra, students' understanding of these concepts is vital to their mastery of the subject. Lay introduces these concepts early in a familiar, concrete Rn setting, develops them gradually, and returns to them again and again throughout the text. Finally, when discussed in the abstract, these concepts are more accessible.

**New To This Edition :**

- New full color design allows for clearer understanding of figures and graphically important concepts and procedures.
- Section 1.6, Applications of Linear Systems, is new to this edition. It includes applications on the Leontief Economic Model, Solving Chemical Equations, and Traffic Flow.
- Section 2.9 from the Updated 2/e, Subspaces of R n, has been split into two sections, 2.8 and 2.9. This forms a better bridge to Chapter 5 for those instructors who wish to cover Eigenvalues earlier in the course, or postpone or omit vector spaces. This option is ideal for a shorter course.
- A new Instructor's Solutions Manual contains detailed solutions for all exercises, including problem-specific teaching notes.
- MyMathLab is now available, integrating the text's content with the Student Study Guide. All of the text's many electronic resources can be found on MyMathLab.
- MathXL is now available for the text, allowing students to take tests and quizzes online.
- An electronic test generator, TestGen-EQ, allows for creation of paper tests, or can be used in conjunction with MathXL to test or do homework online.

**Features :**

- MyMathLab
- Interactive Content. The full text is now available online, integrating the text's content with the Student's Study Guide and electronic practice problems. All of the text's many electronic resources can be found on MyMathLab.
- MathXL allows students to take tests and quizzes online. It works in conjunction with TestGen-EQ, allowing instructors to post custom tests for quizzing, homework, or practice.
- Student's Study Guide is now available free as a part of MyMathLab.
- Transparency Masters for every chapter help save time. For instructors only.
- Downloadable Data Files for exercises and projects are available for Matlab, Maple, Mathematica, and selected graphing calculators.
- Sample Syllabi written by David C. Lay offer suggestions for course organization. For instructors only.
- Review Sheets and Practice Tests help students study effectively.
- Sample Tests with solutions written by David C. Lay provide classroom-tested exam options for instructors. For instructors only.
- Applications by Chapter are a selection of case studies, problems, and application projects based on real-world data and keyed to the text.
- References to Applications lists hundreds of applications referred to in the text, and provides hot links to the case studies and application projects.
- Teaching Notes for Linear Algebra: Modules for Interactive Learning Using Maple®, is an interactive linear algebra tutorial. These highly geometric modules can be used for classroom demonstrations, individual student work or interactive collaboration. For instructors only.
- Fundamental ideas of linear algebra are introduced within the first seven lectures, in the concrete setting of R n, and then gradually examined from different points of view. Later generalizations of these concepts appear as natural extensions of familiar ideas.
- Focus on visualization of concepts throughout the book.
- Icons in the margins to flag topics for which expanded or enhanced material is available on the Web.
- A modern view of matrix multiplication is presented. Definitions and proofs focus on the columns of a matrix rather than on the matrix entries.
- Numerical Notes give a realistic flavor to the text. Students are reminded frequently of issues that arise in the real-life use of linear algebra.
- Each major concept in the course is given a geometric interpretation because many students learn better when they can visualize an idea.
- M exercises appear in every section. To be solved with the aid of a Matrix program such as MATLAB, Maple, Mathematica, MathCad, Derive or programmable calculators with matrix capabilities, such as the TI-83 Plus, TI-86, TI-89, and HP-48G. Data for these exercises are provided on the Web.

**Lay, David C. : University of Maryland**

David C. Lay holds a B.A. from Aurora University (Illinois), and an M.A. and Ph.D. from the University of California at Los Angeles. Lay has been an educator and research mathematician since 1966, mostly at the University of Maryland, College Park. He has also served as a visiting professor at the University of Amsterdam, the Free University in Amsterdam, and the University of Kaiserslautern, Germany. He has over 30 research articles published in functional analysis and linear algebra.

As a founding member of the NSF-sponsored Linear Algebra Curriculum Study Group, Lay has been a leader in the current movement to modernize the linear algebra curriculum. Lay is also co-author of several mathematics texts, including Introduction to Functional Analysis, with Angus E. Taylor, Calculus and Its Applications, with L.J. Goldstein and D.I. Schneider, and Linear Algebra Gems-Assets for Undergraduate Mathematics, with D. Carlson, C.R. Johnson, and A.D. Porter.

A top-notch educator, Professor Lay has received four university awards for teaching excellence, including, in 1996, the title of Distinguished Scholar-Teacher of the University of Maryland. In 1994, he was given one of the Mathematical Association of America's Awards for Distinguished College or Unviersity Teaching of Mathematics. He has been elected by the university students to membership in Alpha Lambda Delta National Scholastic Honor Society and Golden Key National Honor Society. In 1989, Aurora University conferred on him the Outstanding Alumnus award. Lay is a member of the American Mathematical Society, the Canadian Mathematical Society, the International Linear Algebra Society, the Mathematical Association of America, Sigma Xi, and the Society for Industrial and Applied Mathematics. Since 1992, he has served several terms on the national board of the Association of Christians in the Mathematical Sciences.

*(Supplementary Exercises are featured at the end of each chapter.) *

1. Linear Equations in Linear Algebra.

Introductory Example: Linear Models in Economics and Engineering.

Systems of Linear Equations.

Row Reduction and Echelon Forms.

Vector Equations.

The Matrix Equation Ax = b.

Solution Sets of Linear Systems.

Applications of Linear Systems.

Linear Independence.

Introduction to Linear Transformations.

The Matrix of a Linear Transformation.

Linear Models in Business, Science, and Engineering.

2. Matrix Algebra.

Introductory Example: Computer Graphics in Aircraft Design.

Matrix Operations.

The Inverse of a Matrix.

Characterizations of Invertible Matrices.

Partitioned Matrices.

Matrix Factorizations.

The Leontief Input-Output Model.

Applications to Computer Graphics.

Subspaces of R n.

Dimensions and Rank.

3. Determinants.

Introductory Example: Determinants in Analytic Geometry.

Introduction to Determinainants.

Properties of Determinants.

Cramer's Rule, Volume, and Linear Transformations.

4. Vector Spaces.

Introductory Example: Space Flight and Control Systems.

Vector Spaces and Subspaces.

Null Spaces, Column Spaces, and Linear Transformations.

Linearly Independent Sets; Bases.

Coordinate Systems.

The Dimension of Vector Space

Rank.

Change of Basis.

Applications to Difference Equations.

Applications to Markov Chains.

5. Eigenvalues and Eigenvectors.

Introductory Example: Dynamical Systems and Spotted Owls.

Eigenvectors and Eigenvalues.

The Characteristic Equation.

Diagonalization.

Eigenvectors and Linear Transformations.

Complex Eigenvalues.

Discrete Dynamical Systems.

Applications to Differential Equations.

Iterative Estimates for Eigenvalues.

6. Orthogonality and Least-Squares.

Introductory Example: Readjusting the North American Datum.

Inner Product, Length, and Orthogonality.

Orthogonal Sets.

Orthogonal Projections.

The Gram-Schmidt Process.

Least-Squares Problems.

Applications to Linear Models.

Inner Product Spaces.

Applications of Inner Product Spaces.

7. Symmetric Matrices and Quadratic Forms.

Introductory Example: Multichannel Image Processing.

Diagonalization of Symmetric Matrices.

Quadratic Forms.

Constrained Optimization.

The Singular Value Decomposition.

Applications to Image Processing and Statistics.

Appendices.

A. Uniqueness of the Reduced Echelon Form.

B. Complex Numbers

Glossary.

Answers.

Index.