Elementary Linear Algebra 11th edition, by David C. Lay shows how all the essential elements of elementary linear algebra can be derived, using minimum mathematical machinery and concepts, with maximum use of concrete examples and non-technical language. It is intended for students who have had at least one year (or course equivalent) of calculus. The student who has taken calculus but has not seen matrix methods should not feel discouraged from beginning this book, as considerable review will be included; similarly a student whose mathematics education has been purely abstract may be stimulated to see that much of the beauty and usefulness in linear algebra comes from many concise and simple theorems which are easy to state, because of the concrete nature from which they come

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Elementary Linear Algebra 11th edition gives an elementary treatment of linear algebra that is suitable for a first course for undergraduate students. The aim is to present the fundamentals of linear algebra in the clearest possible way; pedagogy is the main consideration. Calculus is not a prerequisite, but there are clearly labeled exercises and examples (which can be omitted without loss of continuity) for students who have studied calculus.

C H A P T E R 1 Systems of Linear Equations and Matrices

1.1 Introduction to Systems of Linear Equations

1.2 Gaussian Elimination

1.3 Matrices and Matrix Operations

1.4 Inverses; Algebraic Properties of Matrices

1.5 Elementary Matrices and a Method for Finding A−1

1.6 More on Linear Systems and Invertible Matrices

1.7 Diagonal, Triangular, and Symmetric Matrices

1.8 Matrix Transformations

1.9 Applications of Linear Systems

• Network Analysis (Traffic Flow)

• Electrical Circuits

• Balancing Chemical Equations

• Polynomial Interpolation

1.10 Application: Leontief Input-Output Models

C H A P T E R 2 Determinants

2.1 Determinants by Cofactor Expansion

2.2 Evaluating Determinants by Row Reduction

2.3 Properties of Determinants; Cramer’s Rule

C H A P T E R 3 Euclidean Vector Spaces

3.1 Vectors in 2-Space, 3-Space, and n-Space

3.2 Norm, Dot Product, and Distance in Rn

3.3 Orthogonality

3.4 The Geometry of Linear Systems

3.5 Cross Product

C H A P T E R 4 General Vector Spaces

4.1 Real Vector Spaces

4.2 Subspaces

4.3 Linear Independence

4.4 Coordinates and Basis

4.5 Dimension

4.6 Change of Basis

4.7 Row Space, Column Space, and Null Space

4.8 Rank, Nullity, and the Fundamental Matrix Spaces

4.9 Basic Matrix Transformations in R2 and R3

4.10 Properties of Matrix Transformations

4.11 Application: Geometry of Matrix Operators on R2

C H A P T E R 5 Eigenvalues and Eigenvectors

5.1 Eigenvalues and Eigenvectors

5.2 Diagonalization

5.3 Complex Vector Spaces

5.4 Application: Differential Equations

5.5 Application: Dynamical Systems and Markov Chains

C H A P T E R 6 Inner Product Spaces

6.1 Inner Products

6.2 Angle and Orthogonality in Inner Product Spaces

6.3 Gram–Schmidt Process; QR-Decomposition

6.4 Best Approximation; Least Squares

6.5 Application: Mathematical Modeling Using Least Squares

6.6 Application: Function Approximation; Fourier Series

C H A P T E R 7 Diagonalization and Quadratic Forms

7.1 Orthogonal Matrices

7.2 Orthogonal Diagonalization

7.5 Hermitian, Unitary, and Normal Matrices

C H A P T E R 8 General Linear Transformations

8.1 General Linear Transformation

8.2 Compositions and Inverse Transformations

8.3 Isomorphism

8.4 Matrices for General Linear Transformations

8.5 Similarity

C H A P T E R 9 Numerical Methods

9.1 LU-Decompositions

9.2 The Power Method

9.3 Comparison of Procedures for Solving Linear Systems

9.4 Singular Value Decomposition

9.5 Application: Data Compression Using Singular Value Decomposition

A P P E N D I X A Working with Proofs

A P P E N D I X B Complex Numbers