A Modern Introduction to Linear Algebra
Purchasing Options
Features
- Develops key concepts in the context of Euclidean n-space
- Explains the theory of matrices
- Explores the differences between finite- and infinite-dimensional vector spaces
- Covers the algebra and matrix representation of linear transformations
- Presents proofs for nearly all results
- Includes a host of examples and various applications reflecting some of the many disciplines that use linear algebra
Pedagogical Features
- Follows the recommendations for a first course in linear algebra provided by the Linear Algebra Curriculum Study Group (LACSG), Augmenting the Teaching of Linear Algebra through the use of Software Tools (ATLAST) project, and the Linear Algebra Modules Project (LAMP)
- Contains numerous exercises of varying levels of difficulty
- Reviews basic mathematical tools in the appendices
Solutions manual available for qualifying instructors
Summary
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
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.
Author Bio(s)
Editorial 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
