A Modern Introduction to Linear Algebra: 1st Edition (Hardback) book cover

A Modern Introduction to Linear Algebra

1st Edition

By Henry Ricardo

Chapman and Hall/CRC

670 pages | 83 B/W Illus.

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Hardback: 9781439800409
pub: 2009-10-21
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Description

Useful Concepts and Results at the Heart of Linear Algebra

A one- or two-semester course for a wide variety of students at the sophomore/junior undergraduate level

A Modern Introduction to Linear Algebra provides a rigorous yet accessible matrix-oriented introduction to the essential concepts of linear algebra. Concrete, easy-to-understand examples motivate the theory.

The book first discusses vectors, Gaussian elimination, and reduced row echelon forms. It then offers a thorough introduction to matrix algebra, including defining the determinant naturally from the PA=LU factorization of a matrix. The author goes on to cover finite-dimensional real vector spaces, infinite-dimensional spaces, linear transformations, and complex vector spaces. The final chapter presents Hermitian and normal matrices as well as quadratic forms.

Taking a computational, algebraic, and geometric approach to the subject, this book provides the foundation for later courses in higher mathematics. It also shows how linear algebra can be used in various areas of application. Although written in a "pencil and paper" manner, the text offers ample opportunities to enhance learning with calculators or computer usage.

Solutions manual available for qualifying instructors

Reviews

"This work is a sound presentation of linear algebra. … Each topic is carefully and thoroughly covered via the pedagogy … The volume includes more than 1,200 exercises, some to be completed manually and others intended to be solved using a computer algebra system. … The generality of approach makes this work appropriate for students in virtually any discipline. Summing Up: Recommended."

CHOICE, June 2010

"The author of this text, Henry Ricardo, has identified several shortcomings of typical courses on linear algebra and provides an exciting offering how to overcome them … The key advantage of this text is that the crucial topic of eigenvalues and eigenvectors is reached as fast as possible! Chapter 4 is earlier than in most texts, and the integration of determinants within that chapter is a good way to keep that topic in the right place today. The single most exciting choice of this text is to start the semester with material on vectors (n-dimensional real vectors). This allows for fast introduction to material that is new to students both to catch their interest and to demonstrate that this class deals with material that is very new to most of them. And it sets up the entire text for the proper perspective in higher level mathematics of having vectors as elements of spaces."

—Matthias Gobbert, University of Maryland, Baltimore County, USA

Table of Contents

Vectors

Vectors in Rn

The Inner Product and Norm

Spanning Sets

Linear Independence

Bases

Subspaces

Summary

Systems of Equations

The Geometry of Systems of Equations in R2 and R3

Matrices and Echelon Form

Gaussian Elimination

Computational Considerations—Pivoting

Gauss–Jordan Elimination and Reduced Row Echelon Form

Ill-Conditioned Systems of Linear Equations

Rank and Nullity of a Matrix

Systems of m Linear Equations in n Unknowns

Matrix Algebra

Addition and Subtraction of Matrices

Matrix–Vector Multiplication

The Product of Two Matrices

Partitioned Matrices

Inverses of Matrices

Elementary Matrices

The LU Factorization

Eigenvalues, Eigenvectors, and Diagonalization

Determinants

Determinants and Geometry

The Manual Calculation of Determinants

Eigenvalues and Eigenvectors

Similar Matrices and Diagonalization

Algebraic and Geometric Multiplicities of Eigenvalues

The Diagonalization of Real Symmetric Matrices

The Cayley–Hamilton Theorem (a First Look)/the Minimal Polynomial

Vector Spaces

Vector Spaces

Subspaces

Linear Independence and the Span

Bases and Dimension

Linear Transformations

Linear Transformations

The Range and Null Space of a Linear Transformation

The Algebra of Linear Transformations

Matrix Representation of a Linear Transformation

Invertible Linear Transformations

Isomorphisms

Similarity

Similarity Invariants of Operators

Inner Product Spaces

Complex Vector Spaces

Inner Products

Orthogonality and Orthonormal Bases

The Gram–Schmidt Process

Unitary Matrices and Orthogonal Matrices

Schur Factorization and the Cayley–Hamilton Theorem

The QR Factorization and Applications

Orthogonal Complements

Projections

Hermitian Matrices and Quadratic Forms

Linear Functionals and the Adjoint of an Operator

Hermitian Matrices

Normal Matrices

Quadratic Forms

Singular Value Decomposition

The Polar Decomposition

Appendix A: Basics of Set Theory

Appendix B: Summation and Product Notation

Appendix C: Mathematical Induction

Appendix D: Complex Numbers

Answers/Hints to Odd-Numbered Problems

Index

A Summary appears at the end of each chapter.

About the Author

Henry Ricardo is a professor of mathematics at Medgar Evers College of the City University of New York, where he was presented with the 2008 Distinguished Service Award by the School of Science, Health and Technology. Dr. Ricardo was also given the 2009 Distinguished Service Award by the Metropolitan New York Section of the MAA, of which he is the Governor.

Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General
MAT020000
MATHEMATICS / Mensuration