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 16 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 79 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 1012 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.
Algebraic Structures.- Matrices and Systems of Linear Equations.- Vector
Spaces.- Linear Transformations.- Dual Spaces.- Inner Product Spaces.-
Canonical Forms of an Operator.- Bilinear and Quadratic Forms.- Sesquilinear
and Hermitian Forms.- Applications of Linear Algebra to Numerical Methods.-
Affine and Euclidean Spaces and the Applications of Linear Algebra to
Geometry.- Ordinary differential equations and linear systems of
ordinary 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 the year 1986. After completing his Ph.D., he started his teaching career as Lecturer at the Department of Mathematics, Aligarh Muslim University, elevated to the post of Reader in 1987 and then became Professor in 2005. He also served as Associate Professor at the Department of Mathematics, King Abdulaziz University, KSA, from 1998 to 2004.
His research interests include ring theory/commutativity and structure of rings and near-rings, derivations on rings, near-rings & Banach algebras, differential identities in rings and algebras, applied linear algebra, algebraic coding theory and cryptography. With a teaching experience of around 35 years, Prof. Ashraf has supervised the Ph.D. thesis of 13 students and is currently guiding 6 more. He has published around 225 research articles in internationaljournals and conference proceedings of repute. He received the Young Scientist's Award from Indian Science Congress Association in the year 1988 and the I.M.S. Prize from Indian Mathematical Society for the year 1995. He has completed many major research projects from the UGC, DST and NBHM. He is also Editor/ Managing Editor of many reputed international mathematical journals.
Vincenzo De Filippis is Associate Professor of Algebra at the University of Messina, Italy. He completed his Ph.D. in Mathematics from the University of Messina, Italy, in 1999. He is the member of the Italian Mathematical Society (UMI) and National Society of Algebraic and Geometric Structures and their Applications (GNSAGA). He has published around 100 research articles in reputed journals and conference proceedings.
Mohammad Aslam Siddeeque is Associate 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 entitled On derivations and related mappings in rings and near-rings". His research interest lies in derivations and its various generalizations on rings and near-rings, on which he has published articles in reputed journals.
