Advanced Linear Algebra with Applications

This is a comprehensive textbook of linear algebra, intended for advanced undergraduate students of mathematics. Throughout the text, proofs have been made direct and simple without irrelevant details. It needs no prerequisites except some basic knowledge of sets, mappings and number systems. The topics have been presented in a simple and coherent style. The book starts with a quick review of basic literature on groups, rings and fields and ends with the applications of linear algebra in analytical geometry and numerical methods. It also includes a number of exercises and examples at the end of each chapter to enhance the understanding of readers. The subject is developed in a manner accessible to students with little knowledge of linear algebra.

Chapters 1-6 discuss introductory topics like groups, rings, fields, matrices, determinants, systems of linear equations, vector spaces, linear transformations, dual spaces, and inner product spaces, which are taught at the first course of linear algebra at the undergraduate level. Chapters 7-9 cover advanced topics like canonical, bilinear, quadratic, sesquilinear and Hermitian forms of operators and matrices, which are taught at the advanced undergraduate course in mathematics. Chapters 10-12 focus on empowering readers to pursue interdisciplinary applications of linear algebra and illustrate the power of the subject through a variety of applications in numerical methods, analytical geometry and in solving linear system of differential equations.

MOHAMMAD ASHRAF is Professor at the Department of Mathematics, Aligarh Muslim University, India. He completed his Ph.D. in Mathematics from Aligarh Muslim University, India, in 1986, with the thesis titled "A study of certain commutativity conditions for associative rings". He has authored over 200 research publications in various areas of mathematics such as ring theory, coding theory and graph theory. He is a recipient of the Young Scientist's Award (1988) from Indian Science Congress Association, I.M.S. Prize (1995) from Indian Mathematical Society, and the Ziauddin Gold Medal for securing first position at M.Sc. examination in the year 1981. He is presently dealing with two major research projects awarded by the Department of Science and Technology in the area of derivations in rings. His research interests include ring theory/commutativity and structure of rings and near-rings, derivations on rings, near-rings and Banach algebras, functional identities in rings and algebras, applied linear algebra, algebraic coding theory, cryptography, and graph theory.

VINCENZO DE FILIPPIS is Associate Professor at the Department of Mathematics, University of Messina, Italy. He completed his Ph.D. in Mathematics from the University of Messina, Italy, in 1999 with the thesis titled "Leads in rings first". His research articles have been published in journals and conference proceedings of repute. He is the recipient of the Young Researcher Prize by the University of Messina, Italy, in 2005.

MOHAMMAD ASLAM SIDDEEQUE is Assistant Professor at the Department of Mathematics, Aligarh Muslim University, India. He completed his Ph.D. in Mathematics from Aligarh Muslim University, India, in 2014 with the thesis titled "On derivations and related mappings in rings and near-rings". His area of research includes the study of derivations over rings and near-rings on which he has published research articles in journals of repute.