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by David C. Lay

Edition: 2ND 97Copyright: 1997

Publisher: Addison-Wesley Longman, Inc.

Published: 1997

International: No

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Linear algebra is relatively easy for students when the material is presented in a familiar, concrete setting, but it becomes much more difficult when abstract concepts are introduced. Certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations) are not easily understood but are fundamental to the study of linear algebra. The author introduces these concepts early in a familiar, concrete R^{n} setting, develops them gradually, and returns to them again and again throughout the text. Finally, when discussed in the abstract, these concepts more accessible and student understanding is reinforced through True-or-False questions, practice problems, and the use of technologies, such as MATLAB, for solving large matrix exercises.

**FEATURES:**

- Technology exercises, using real-data sets and citing sources, provide realistic exercises designed to be completed with technology.
- True-or-False questions in the exercise sets encourage critical thinking and discussion.
- To enhance clarity, 25 percent of the art has been revised.
- This edition includes new and revised Numerical Notes, some of which address MAPLE.
- Chapters 1 and 2 of the first edition were combined and streamlined to allow for a smoother entry into the course.
- Early coverage of Subspaces of R
^{n}in new Section 2.9 lets users skip the bulk of material in Chapters 3 and 4 and go straight to the Eigenvalue section in Chapter 5. Those teaching a standard-paced course would cover Chapters 2, 3, and 4, skipping 2.9. - LU Factorization coverage is emphasized.
- Difference equations, particularly important to a junior-level course, have been added to Chapter 6.
- This text includes such up-to-date topics as difference equations (dynamical systems); LU and QR factorizations; and spectoral decomposition.
- The Application section at the end of each chapter provide a broad selection of applications from engineering, physics, computer science, mathematics, economics, and statistics.
- Practice Problems & Answers, unique to this text, focus on points that may cause students difficulty. They are placed before the section exercise sets as a warmup; answers follow the exercise sets.
- The Supplemental Exercises following each chapter include theoretical and proof-oriented multiconcept stretch exercises and writing exercises that require written justification of a true-or-false answer.
- A glossary of key terms is included.
- Numerical Notes, designed to make the reader more computer-aware, are given throughout the text, indicating technology-related issues that deserve further study.

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

**Chapter 1: Linear Equations in Linear Algebra**

Systems of Linear Equations

Row Reduction and Echelon Forms

Vector Equations

The Matrix Equation Ax = b

Solution Sets of Linear Systems

Linear Independence

Introduction to Linear Transformations

The Matrix of a Linear Transformation

Linear Models in Business, Science, and Engineering

**Chapter 2: Matrix Algebra**

Matrix Operations

The Inverse of a Matrix

Characterizations of Invertible Matrices

Partitioned Matrices

Matrix Factorizations

Iterative Solutions of Linear Systems

The Leontief InputOutput Model

Applications to Computer Graphics

Subspaces of R^{n}

**Chapter 3: Determinants**

Introduction to Determinants

Properties of Determinants

Cramer's Rule, Volume, and Linear Transformations

**Chapter 4: Vector Spaces**

Vector Spaces and Subspaces

Null Spaces, Column Spaces, and Linear Transformations

Linearly Independent Sets; Bases

Coordinate Systems

The Dimension of a Vector Space

Rank

Change of Basis

Applications to Difference Equations

Applications to Markov Chains

**Chapter 5: Eigenvalues and Eigenvectors**

Eigenvectors and Eigenvalues

The Characteristic Equation

Diagonalization

Eigenvectors and Linear Transformations

Complex Eigenvalues

Discrete Dynamical Systems

Applications to Differential Equations

Iterative Estimates for Eigenvalues

**Chapter 6: Orthogonality and Least-Squares**

Inner Product, Length, and Orthogonality

Orthogonal Sets

Orthogonal Projections

The GramSchmidt Process

Least-Squares Problems

Applications to Linear Models

Inner Product Spaces

Applications of Inner Product Spaces

**Chapter 7: Symmetric Matrices and Quadratic Forms**

Diagonalization of Symmetric Matrices

Quadratic Forms

Constrained Optimization

The Singular Value Decomposition

Applications to Image Processing and Statistics

Appendices

Uniqueness of the Reduced Echelon Form

Complex Numbers

Summary

Linear algebra is relatively easy for students when the material is presented in a familiar, concrete setting, but it becomes much more difficult when abstract concepts are introduced. Certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations) are not easily understood but are fundamental to the study of linear algebra. The author introduces these concepts early in a familiar, concrete R^{n} setting, develops them gradually, and returns to them again and again throughout the text. Finally, when discussed in the abstract, these concepts more accessible and student understanding is reinforced through True-or-False questions, practice problems, and the use of technologies, such as MATLAB, for solving large matrix exercises.

**FEATURES:**

- Technology exercises, using real-data sets and citing sources, provide realistic exercises designed to be completed with technology.
- True-or-False questions in the exercise sets encourage critical thinking and discussion.
- To enhance clarity, 25 percent of the art has been revised.
- This edition includes new and revised Numerical Notes, some of which address MAPLE.
- Chapters 1 and 2 of the first edition were combined and streamlined to allow for a smoother entry into the course.
- Early coverage of Subspaces of R
^{n}in new Section 2.9 lets users skip the bulk of material in Chapters 3 and 4 and go straight to the Eigenvalue section in Chapter 5. Those teaching a standard-paced course would cover Chapters 2, 3, and 4, skipping 2.9. - LU Factorization coverage is emphasized.
- Difference equations, particularly important to a junior-level course, have been added to Chapter 6.
- This text includes such up-to-date topics as difference equations (dynamical systems); LU and QR factorizations; and spectoral decomposition.
- The Application section at the end of each chapter provide a broad selection of applications from engineering, physics, computer science, mathematics, economics, and statistics.
- Practice Problems & Answers, unique to this text, focus on points that may cause students difficulty. They are placed before the section exercise sets as a warmup; answers follow the exercise sets.
- The Supplemental Exercises following each chapter include theoretical and proof-oriented multiconcept stretch exercises and writing exercises that require written justification of a true-or-false answer.
- A glossary of key terms is included.
- Numerical Notes, designed to make the reader more computer-aware, are given throughout the text, indicating technology-related issues that deserve further study.

Author Bio

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

Table of Contents

**Chapter 1: Linear Equations in Linear Algebra**

Systems of Linear Equations

Row Reduction and Echelon Forms

Vector Equations

The Matrix Equation Ax = b

Solution Sets of Linear Systems

Linear Independence

Introduction to Linear Transformations

The Matrix of a Linear Transformation

Linear Models in Business, Science, and Engineering

**Chapter 2: Matrix Algebra**

Matrix Operations

The Inverse of a Matrix

Characterizations of Invertible Matrices

Partitioned Matrices

Matrix Factorizations

Iterative Solutions of Linear Systems

The Leontief InputOutput Model

Applications to Computer Graphics

Subspaces of R^{n}

**Chapter 3: Determinants**

Introduction to Determinants

Properties of Determinants

Cramer's Rule, Volume, and Linear Transformations

**Chapter 4: Vector Spaces**

Vector Spaces and Subspaces

Null Spaces, Column Spaces, and Linear Transformations

Linearly Independent Sets; Bases

Coordinate Systems

The Dimension of a Vector Space

Rank

Change of Basis

Applications to Difference Equations

Applications to Markov Chains

**Chapter 5: Eigenvalues and Eigenvectors**

Eigenvectors and Eigenvalues

The Characteristic Equation

Diagonalization

Eigenvectors and Linear Transformations

Complex Eigenvalues

Discrete Dynamical Systems

Applications to Differential Equations

Iterative Estimates for Eigenvalues

**Chapter 6: Orthogonality and Least-Squares**

Inner Product, Length, and Orthogonality

Orthogonal Sets

Orthogonal Projections

The GramSchmidt Process

Least-Squares Problems

Applications to Linear Models

Inner Product Spaces

Applications of Inner Product Spaces

**Chapter 7: Symmetric Matrices and Quadratic Forms**

Diagonalization of Symmetric Matrices

Quadratic Forms

Constrained Optimization

The Singular Value Decomposition

Applications to Image Processing and Statistics

Appendices

Uniqueness of the Reduced Echelon Form

Complex Numbers

Publisher Info

Publisher: Addison-Wesley Longman, Inc.

Published: 1997

International: No

Published: 1997

International: No

Linear algebra is relatively easy for students when the material is presented in a familiar, concrete setting, but it becomes much more difficult when abstract concepts are introduced. Certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations) are not easily understood but are fundamental to the study of linear algebra. The author introduces these concepts early in a familiar, concrete R^{n} setting, develops them gradually, and returns to them again and again throughout the text. Finally, when discussed in the abstract, these concepts more accessible and student understanding is reinforced through True-or-False questions, practice problems, and the use of technologies, such as MATLAB, for solving large matrix exercises.

**FEATURES:**

- Technology exercises, using real-data sets and citing sources, provide realistic exercises designed to be completed with technology.
- True-or-False questions in the exercise sets encourage critical thinking and discussion.
- To enhance clarity, 25 percent of the art has been revised.
- This edition includes new and revised Numerical Notes, some of which address MAPLE.
- Chapters 1 and 2 of the first edition were combined and streamlined to allow for a smoother entry into the course.
- Early coverage of Subspaces of R
^{n}in new Section 2.9 lets users skip the bulk of material in Chapters 3 and 4 and go straight to the Eigenvalue section in Chapter 5. Those teaching a standard-paced course would cover Chapters 2, 3, and 4, skipping 2.9. - LU Factorization coverage is emphasized.
- Difference equations, particularly important to a junior-level course, have been added to Chapter 6.
- This text includes such up-to-date topics as difference equations (dynamical systems); LU and QR factorizations; and spectoral decomposition.
- The Application section at the end of each chapter provide a broad selection of applications from engineering, physics, computer science, mathematics, economics, and statistics.
- Practice Problems & Answers, unique to this text, focus on points that may cause students difficulty. They are placed before the section exercise sets as a warmup; answers follow the exercise sets.
- The Supplemental Exercises following each chapter include theoretical and proof-oriented multiconcept stretch exercises and writing exercises that require written justification of a true-or-false answer.
- A glossary of key terms is included.
- Numerical Notes, designed to make the reader more computer-aware, are given throughout the text, indicating technology-related issues that deserve further study.

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

**Chapter 1: Linear Equations in Linear Algebra**

Systems of Linear Equations

Row Reduction and Echelon Forms

Vector Equations

The Matrix Equation Ax = b

Solution Sets of Linear Systems

Linear Independence

Introduction to Linear Transformations

The Matrix of a Linear Transformation

Linear Models in Business, Science, and Engineering

**Chapter 2: Matrix Algebra**

Matrix Operations

The Inverse of a Matrix

Characterizations of Invertible Matrices

Partitioned Matrices

Matrix Factorizations

Iterative Solutions of Linear Systems

The Leontief InputOutput Model

Applications to Computer Graphics

Subspaces of R^{n}

**Chapter 3: Determinants**

Introduction to Determinants

Properties of Determinants

Cramer's Rule, Volume, and Linear Transformations

**Chapter 4: Vector Spaces**

Vector Spaces and Subspaces

Null Spaces, Column Spaces, and Linear Transformations

Linearly Independent Sets; Bases

Coordinate Systems

The Dimension of a Vector Space

Rank

Change of Basis

Applications to Difference Equations

Applications to Markov Chains

**Chapter 5: Eigenvalues and Eigenvectors**

Eigenvectors and Eigenvalues

The Characteristic Equation

Diagonalization

Eigenvectors and Linear Transformations

Complex Eigenvalues

Discrete Dynamical Systems

Applications to Differential Equations

Iterative Estimates for Eigenvalues

**Chapter 6: Orthogonality and Least-Squares**

Inner Product, Length, and Orthogonality

Orthogonal Sets

Orthogonal Projections

The GramSchmidt Process

Least-Squares Problems

Applications to Linear Models

Inner Product Spaces

Applications of Inner Product Spaces

**Chapter 7: Symmetric Matrices and Quadratic Forms**

Diagonalization of Symmetric Matrices

Quadratic Forms

Constrained Optimization

The Singular Value Decomposition

Applications to Image Processing and Statistics

Appendices

Uniqueness of the Reduced Echelon Form

Complex Numbers