Introduction to Linear Algebra

Introduction to Linear Algebra, 5/e is a foundation book that bridges both practical computation and theoretical principles. Due to its flexible table of contents, the book is accessible for both students majoring in the scientific, engineering, and social sciences, as well as students that want an introduction to mathematical abstraction and logical reasoning. In order to achieve the text's flexibility, the book centers on 3 principal topics: matrix theory and systems of linear equations, elementary vector space concepts, and the eigenvalue problem. This highly adaptable text can be used for a one-quarter or one-semester course at the sophomore/junior level, or for a more advanced class at the junior/senior level.

Appropriate Courses
Introductory Linear Algebra.

Features of the Fifth Edition

NEW Chapter—A new Chapter 2, “Vectors in 2-space and 3-Space” makes a smoother introduction to the ideas of vector space, and allows for a more geometric emphasis in the text.
An early introduction to vector space ideas—In Chapter 3, elementary vector space ideas (subspace, basis, dimension, and so on) are introduced in the familiar setting of R^n.
An early introduction to eigenvalues—It is now possible with this text to cover the eigenvalue problem very early and in much greater depth. A brief introduction to determinants is given in Section 4.2 to facilitate the early treatment of eigenvalues.
An early introduction to linear combinations—In Section 1.5, the matrix-vector product Ax is expressed as a linear combination of the columns of A, Ax = x1A1 + x2A2 + . . . + xnAn. This viewpoint leads to a simple and natural development for the theory associated with systems of linear equations. This approach gives some early motivation for the vector space concepts (introduced in Chapter 3) such as subspace, basis, and dimension.
Applications to different fields of study—Provides motivation for students in a wide variety of disciplines.

Hallmark Features

• A gradual increase in the level of difficulty. In a typical linear algebra course, students find the techniques of Gaussian elimination and matrix operations fairly easy. Then, the ensuing material relating to vector spaces is suddenly much harder. The authors have done three things to lessen this abrupt midterm jump in difficulty:
• 1. Introduction of linear independence early, in Section 1.7.
• 2. A new Chapter 2, “Vectors in 2-space and 3-Space.”
• 3 .Introduction of vector space concepts such as subspace, basis and dimension in Chapter 3, in the familiar geometrical setting of R^n.
• Clarity of exposition. For many students, linear algebra is the most rigorous and abstract mathematical course they have taken since high-school geometry. The authors have tried to write the text so that it is accessible, but also so that it reveals something of the power of mathematical abstraction. To this end, the topics have been organized so that they flow logically and naturally from the concrete and computational to the more abstract. Numerous examples, many presented in extreme detail, have been included in order to illustrate the concepts.
• Supplementary exercises. A set of supplementary exercises are included at the end of each chapter. These exercises, some of which are true-false questions, are designed to test the student's understanding of important concepts. They often require the student to use ideas from several different sections.
• Extensive exercise sets. Numerous exercises, ranging from routine drill exercises to interesting applications, and exercises of a theoretical nature. The more difficult theoretical exercises have fairly substantial hints. The computational exercises are written using workable numbers that do not obscure the point with a mass of cumbersome arithmetic details.
• Spiraling exercises. Many sections contain a few exercises that hint at ideas that will be developed later.
• Historical notes. These assist the student in gaining a historical and mathematical perspective of the ideas and concepts of linear algebra.
• Integration of MATLAB. We have included a collection of MATLAB projects at the end of each chapter. For the student who is interested in computation, these projects provide hands-on experience with MATLAB.
• A short MATLAB appendix. A brief appendix on using MATLAB for problems that typically arise in linear algebra is included.
• New Technology Resource Manual. Support is now included for Maple and Mathematica, as well as MATLAB.

1. Matrices and Systems of Linear Equations.

Introduction to Matrices and Systems of Linear Equations.
Echelon Form and Gauss-Jordan Elimination.
Consistent Systems of Linear Equations.
Applications (Optional).
Matrix Operations.
Algebraic Properties of Matrix Operations.
Linear Independence and Nonsingular Matrices.
Data Fitting, Numerical Integration, and Numerical Differentiation (Optional).
Matrix Inverses and Their Properties.

2. Vectors in 2-Space and 3-Space.

Vectors in the Plane.
Vectors in Space.
The Dot Product and the Cross Product.
Lines and Planes in Space.

3. The Vector Space R^n.

Introduction.
Vector Space Properties of R^n.
Examples of Subspaces.
Bases for Subspaces.
Dimension.
Orthogonal Bases for Subspaces.
Linear Transformations from R^n to R^m.
Least-Squares Solutions to Inconsistent Systems, with Applications to Data Fitting.
Theory and Practice of Least Squares.

4. The Eigenvalue Problem.

The Eigenvalue Problem for (2 x 2) Matrices.
Determinants and the Eigenvalue Problem.
Elementary Operations and Determinants (Optional).
Eigenvalues and the Characteristic Polynomial.
Eigenvectors and Eigenspaces.
Complex Eigenvalues and Eigenvectors.
Similarity Transformations and Diagonalization.
Difference Equations; Markov Chains, Systems of Differential Equations (Optional).

5. Vector Spaces and Linear Transformations.

Introduction.
Vector Spaces.
Subspaces.
Linear Independence, Bases, and Coordinates.
Dimension.
Inner-Product Spaces, Orthogonal Bases, and Projections (Optional).
Linear Transformations.
Operations with Linear Transformations.
Matrix Representations for Linear Transformations.
Change of Basis and Diagonalization.

6. Determinants.

Introduction.
Cofactor Expansions of Determinants.
Elementary Operations and Determinants.
Cramer's Rule.
Applications of Determinants: Inverses and Wronksians.

7. Eigenvalues and Applications.

Quadratic Forms.
Systems of Differential Equations.
Transformation to Hessenberg Form.
Eigenvalues of Hessenberg Matrices.
Householder Transformations.
The QR Factorization and Least-Squares Solutions.
Matrix Polynomials and the Cayley-Hamilton Theorem.
Generalized Eigenvectors and Solutions of Systems of Differential Equations.

Appendix: An Introduction to MATLAB.
Answers to Selected Odd-Numbered Exercises.
Index.