- David C. Lay
- Steven R. Lay
- Judi J. McDonald
- 6th edition | Published by Pearson (July 14th 2021) - Copyright © 2021
- ISBN-13: 9781292351216

*For courses in Linear Algebra.*

**Fosters the concepts and skills needed for future careers**

** Linear Algebra and Its Applications **offers a modern elementary introduction with broad, relevant applications. With traditional texts, the early stages of the course are relatively easy as material is presented in a familiar, concrete setting, but students often hit a wall when abstract concepts are introduced. Certain concepts fundamental to the study of linear algebra (such as linear independence, vector space, and linear transformations) require time to assimilate — and students' understanding of them is vital.

Lay, Lay, and McDonald make these concepts more accessible by introducing them early in a familiar, concrete ℝ* ^{n }*setting, developing them gradually, and returning to them throughout the text so that students can grasp them when they are discussed in the abstract. The

**6th Edition**offers exciting new material, examples, and online resources, along with new topics, vignettes, and applications.

**1. Linear Equations in LinearAlgebra**

Introductory Example: LinearModels in Economics and Engineering

1.1 Systems of Linear Equations

1.2 Row Reduction and EchelonForms

1.3 Vector Equations

1.4 The Matrix Equation*A***x= b**

1.5 Solution Sets of LinearSystems

1.6 Applications of Linear Systems

1.7 Linear Independence

1.8 Introduction to LinearTransformations

1.9 The Matrix of a LinearTransformation

1.10 Linear Models in Business,Science, and Engineering

Projects

Supplementary Exercises

**2. Matrix Algebra**

Introductory Example: ComputerModels in Aircraft Design

2.1 Matrix Operations

2.2 The Inverse of a Matrix

2.3 Characterizations ofInvertible Matrices

2.4 Partitioned Matrices

2.5 Matrix Factorizations

2.6 The Leontief Input—OutputModel

2.7 Applications to ComputerGraphics

2.8 Subspaces ofℝ^{n}

2.9 Dimension and Rank

Projects

Supplementary Exercises

**3. Determinants**

Introductory Example: Random Pathsand Distortion

3.1 Introduction to Determinants

3.2 Properties of Determinants

3.3 Cramer’s Rule, Volume, andLinear Transformations

Projects

Supplementary Exercises

**4. Vector Spaces**

Introductory Example: Space Flightand Control Systems

4.1 Vector Spaces and Subspaces

4.2 Null Spaces, Column Spaces,and Linear Transformations

4.3 Linearly Independent Sets;Bases

4.4 Coordinate Systems

4.5 The Dimension of a VectorSpace

4.6 Change of Basis

4.7 Digital Signal Processing

4.8 Applications to DifferenceEquations

Projects

Supplementary Exercises

**5. Eigenvalues and Eigenvectors**

Introductory Example: DynamicalSystems and Spotted Owls

5.1 Eigenvectors and Eigenvalues

5.2 The Characteristic Equation

5.3 Diagonalization

5.4 Eigenvectors and LinearTransformations

5.5 Complex Eigenvalues

5.6 Discrete Dynamical Systems

5.7 Applications to DifferentialEquations

5.8 Iterative Estimates forEigenvalues

5.9 Markov Chains

Projects

Supplementary Exercises

**6. Orthogonality and Least Squares**

Introductory Example: The NorthAmerican Datum and GPS Navigation

6.1 Inner Product, Length, andOrthogonality

6.2 Orthogonal Sets

6.3 Orthogonal Projections

6.4 The Gram—Schmidt Process

6.5 Least-Squares Problems

6.6 Machine Learning and LinearModels

6.7 Inner Product Spaces

6.8 Applications of Inner ProductSpaces

Projects

Supplementary Exercises

**7. Symmetric Matrices andQuadratic Forms**

Introductory Example: MultichannelImage Processing

7.1 Diagonalization of SymmetricMatrices

7.2 Quadratic Forms

7.3 Constrained Optimization

7.4 The Singular ValueDecomposition

7.5 Applications to ImageProcessing and Statistics

Projects

Supplementary Exercises

**8. The Geometry of Vector Spaces**

Introductory Example: The PlatonicSolids

8.1 Affine Combinations

8.2 Affine Independence

8.3 Convex Combinations

8.4 Hyperplanes

8.5 Polytopes

8.6 Curves and Surfaces

Projects

Supplementary Exercises

**9. Optimization**

Introductory Example: The BerlinAirlift

9.1 Matrix Games

9.2 Linear Programming–GeometricMethod

9.3 Linear Programming–SimplexMethod

9.4 Duality

Projects

Supplementary Exercises

**10. Finite-State Markov Chains(Online Only)**

Introductory Example: GooglingMarkov Chains

10.1 Introduction and Examples

10.2 The Steady-State Vector andGoogle's PageRank

10.3 Communication Classes

10.4 Classification of States andPeriodicity

10.5 The Fundamental Matrix

10.6 Markov Chains and BaseballStatistics

**Appendices**

A. Uniqueness of the ReducedEchelon Form

B. Complex Numbers