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by David C. Lay

Edition: 2ND 00Copyright: 2000

Publisher: Addison-Wesley Longman, Inc.

Published: 2000

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. Students' conceptual understanding is reinforced through True/False questions, practice problems, and the use of technology. David Lay changed the face of linear algebra with the execution of this philosophy, and continues his quest to improve the way linear algebra is taught with the new Updated Second Edition. With this update, he builds on this philosophy through increased visualization in the text, vastly enhanced technology support, and an extensive instructor support package. He has added additional figures to the text to help students visualize abstract concepts at key points in the course. A new dedicated CD and Website further enhance the course materials by providing additional support to help students gain command of difficult concepts. The CD, included in the back of the book, contains a wealth of new materials, with a registration coupon allowing access to a password-protected Website. These new materials are tied directly to the text, providing a comprehensive package for teaching and learning linear algebra.

**1. Linear Equations in Linear Algebra. **

Systems of Linear Equations.

Row Reduction and Echelon Forms.

Vector Equations.

The Matrix Equation *A x = 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.

**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 Input-Output Model.

Applications to Computer Graphics.

Subspaces of R

**3. Determinants. **

Introduction to Determinants.

Properties of Determinants.

Cramer's Rule, Volume, and Linear Transformations.

**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.

**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.

**6. Orthogonality and Least-Squares. **

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. **

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.

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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. Students' conceptual understanding is reinforced through True/False questions, practice problems, and the use of technology. David Lay changed the face of linear algebra with the execution of this philosophy, and continues his quest to improve the way linear algebra is taught with the new Updated Second Edition. With this update, he builds on this philosophy through increased visualization in the text, vastly enhanced technology support, and an extensive instructor support package. He has added additional figures to the text to help students visualize abstract concepts at key points in the course. A new dedicated CD and Website further enhance the course materials by providing additional support to help students gain command of difficult concepts. The CD, included in the back of the book, contains a wealth of new materials, with a registration coupon allowing access to a password-protected Website. These new materials are tied directly to the text, providing a comprehensive package for teaching and learning linear algebra.

Table of Contents

**1. Linear Equations in Linear Algebra. **

Systems of Linear Equations.

Row Reduction and Echelon Forms.

Vector Equations.

The Matrix Equation *A x = 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.

**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 Input-Output Model.

Applications to Computer Graphics.

Subspaces of R

**3. Determinants. **

Introduction to Determinants.

Properties of Determinants.

Cramer's Rule, Volume, and Linear Transformations.

**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.

**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.

**6. Orthogonality and Least-Squares. **

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. **

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: 2000

International: No

Published: 2000

International: No

**1. Linear Equations in Linear Algebra. **

Systems of Linear Equations.

Row Reduction and Echelon Forms.

Vector Equations.

The Matrix Equation *A x = 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.

**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 Input-Output Model.

Applications to Computer Graphics.

Subspaces of R

**3. Determinants. **

Introduction to Determinants.

Properties of Determinants.

Cramer's Rule, Volume, and Linear Transformations.

**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.

**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.

**6. Orthogonality and Least-Squares. **

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. **

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.