Ricardo A Modern Introduction to Linear Algebra

1. Auflage 2012
ISBN: 978-1-4398-9461-3
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 670 Seiten

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

Zielgruppe

Undergraduate students in linear algebra; graduate students and researchers in mathematics.

Autoren/Hrsg.

Ricardo, Henry

Fachgebiete

• Mathematik | Informatik Mathematik Algebra
• Mathematik | Informatik Mathematik Mathematik Allgemein

Inhaltsverzeichnis

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.

Über Autor(innen)

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.