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This book is designed to serve as a textbook for courses offered to
undergraduate and postgraduate students enrolled in Mathematics.
Using elementary row operations and Gram-Schmidt orthogonalization
as basic tools the text develops characterization of equivalence
and similarity, and various factorizations such as rank
factorization, OR-factorization, Schurtriangularization,
Diagonalization of normal matrices, Jordan decomposition, singular
value decomposition, and polar decomposition. Along with
Gauss-Jordan elimination for linear systems, it also discusses best
approximations and least-squares solutions. The book includes norms
on matrices as a means to deal with iterative solutions of linear
systems and exponential of a matrix. The topics in the book are
dealt with in a lively manner. Each section of the book has
exercises to reinforce the concepts, and problems have been added
at the end of each chapter. Most of these problems are theoretical,
and they do not fit into the running text linearly. The detailed
coverage and pedagogical tools make this an ideal textbook for
students and researchers enrolled in senior undergraduate and
beginning postgraduate mathematics courses.
This book introduces the fundamental concepts, techniques and
results of linear algebra that form the basis of analysis, applied
mathematics and algebra. Intended as a text for undergraduate
students of mathematics, science and engineering with a knowledge
of set theory, it discusses the concepts that are constantly used
by scientists and engineers. It also lays the foundation for the
language and framework for modern analysis and its applications.
Divided into seven chapters, it discusses vector spaces, linear
transformations, best approximation in inner product spaces,
eigenvalues and eigenvectors, block diagonalisation,
triangularisation, Jordan form, singular value decomposition, polar
decomposition, and many more topics that are relevant to
applications. The topics chosen have become well-established over
the years and are still very much in use. The approach is both
geometric and algebraic. It avoids distraction from the main theme
by deferring the exercises to the end of each section. These
exercises aim at reinforcing the learned concepts rather than as
exposing readers to the tricks involved in the computation.
Problems included at the end of each chapter are relatively
advanced and require a deep understanding and assimilation of the
topics.
This book introduces the fundamental concepts, techniques and
results of linear algebra that form the basis of analysis, applied
mathematics and algebra. Intended as a text for undergraduate
students of mathematics, science and engineering with a knowledge
of set theory, it discusses the concepts that are constantly used
by scientists and engineers. It also lays the foundation for the
language and framework for modern analysis and its applications.
Divided into seven chapters, it discusses vector spaces, linear
transformations, best approximation in inner product spaces,
eigenvalues and eigenvectors, block diagonalisation,
triangularisation, Jordan form, singular value decomposition, polar
decomposition, and many more topics that are relevant to
applications. The topics chosen have become well-established over
the years and are still very much in use. The approach is both
geometric and algebraic. It avoids distraction from the main theme
by deferring the exercises to the end of each section. These
exercises aim at reinforcing the learned concepts rather than as
exposing readers to the tricks involved in the computation.
Problems included at the end of each chapter are relatively
advanced and require a deep understanding and assimilation of the
topics.
The foundation of computer science is built upon the following
questions: What is an algorithm? What can be computed and what
cannot be computed? What does it mean for a function to be
computable? How does computational power depend upon programming
constructs? Which algorithms can be considered feasible? For more
than 70 years, computer scientists are searching for answers to
such qu- tions. Their ingenious techniques used in answering these
questions form the theory of computation. Theory of computation
deals with the most fundamental ideas of computer s- ence in an
abstract but easily understood form. The notions and techniques
employed are widely spread across various topics and are found in
almost every branch of c- puter science. It has thus become more
than a necessity to revisit the foundation, learn the techniques,
and apply them with con?dence. Overview and Goals This book is
about this solid, beautiful, and pervasive foundation of computer
s- ence. It introduces the fundamental notions, models, techniques,
and results that form the basic paradigms of computing. It gives an
introduction to the concepts and mathematics that computer
scientists of our day use to model, to argue about, and to predict
the behavior of algorithms and computation. The topics chosen here
have shown remarkable persistence over the years and are very much
in current use.
This book is designed to serve as a textbook for courses offered to
undergraduate and postgraduate students enrolled in Mathematics.
Using elementary row operations and Gram-Schmidt orthogonalization
as basic tools the text develops characterization of equivalence
and similarity, and various factorizations such as rank
factorization, OR-factorization, Schurtriangularization,
Diagonalization of normal matrices, Jordan decomposition, singular
value decomposition, and polar decomposition. Along with
Gauss-Jordan elimination for linear systems, it also discusses best
approximations and least-squares solutions. The book includes norms
on matrices as a means to deal with iterative solutions of linear
systems and exponential of a matrix. The topics in the book are
dealt with in a lively manner. Each section of the book has
exercises to reinforce the concepts, and problems have been added
at the end of each chapter. Most of these problems are theoretical,
and they do not fit into the running text linearly. The detailed
coverage and pedagogical tools make this an ideal textbook for
students and researchers enrolled in senior undergraduate and
beginning postgraduate mathematics courses.
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