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.