Advanced Linear Algebra with Applications

Description

This book provides a comprehensive knowledge of linear algebra for graduate and undergraduate courses. As a self-contained text, it aims at covering all important areas of the subject, including algebraic structures, matrices and systems of linear equations, vector spaces, linear transformations, dual and inner product spaces, canonical, bilinear, quadratic, sesquilinear, Hermitian forms of operators and tensor products of vector spaces with their algebras. The last three chapters focus on empowering readers to pursue interdisciplinary applications of linear algebra in numerical methods, analytical geometry and in solving linear system of differential equations. A rich collection of examples and exercises are present at the end of each section to enhance the conceptual understanding of readers. Basic knowledge of various notions, such as sets, relations, mappings, etc., has been pre-assumed. About the Author 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 international journals 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. Preface Contents About the Authors Symbols 1 Algebraic Structures and Matrices 1.1 Groups 1.2 Rings 1.3 Fields with Basic Properties 1.4 Matrices 1.5 System of Linear Equations 2 Vector Spaces 2.1 Definitions and Examples 2.2 Linear Dependence, Independence and Basis 2.3 Geometrical Interpretations 2.4 Change of Basis 3 Linear Transformations 3.1 Kernel and Range of a Linear Transformation 3.2 Basic Isomorphism Theorems 3.3 Algebra of Linear Transformations 3.4 Nonsingular, Singular and Invertible Linear Transformations 3.5 Matrix of a Linear Transformation 3.6 Effect of Change of Bases on a Matrix Representation of a Linear Transformation 4 Dual Spaces 4.1 Linear Functionals and the Dual Space 4.2 Second Dual Space 4.3 Annihilators 4.4 Hyperspaces or Hyperplanes 4.5 Dual (or Transpose) of Linear Transformation 5 Inner Product Spaces 5.1 Inner Products 5.2 The Length of a Vector 5.3 Orthogonality and Orthonormality 5.4 Operators on Inner Product Spaces 6 Canonical Forms of an Operator 6.1 Eigenvalues and Eigenvectors 6.2 Triangularizable Operators 6.3 Diagonalizable Operators 6.4 Jordan Canonical Form of an Operator 6.5 Cayley-Hamilton Theorem and Minimal Polynomial 6.6 Normal Operators on Inner Product Spaces 7 Bilinear and Quadratic Forms 7.1 Bilinear Forms and Their Matrices 7.2 The Effect of the Change of Bases 7.3 Symmetric, Skew-Symmetric and Alternating Bilinear Forms 7.4 Orthogonality and Reflexive Forms 7.5 The Restriction of a Bilinear Form 7.6 Non-degenerate Bilinear Forms 7.7 Diagonalization of Symmetric Forms 7.8 The Orthogonalization Process for Nonisotropic Symmetric Spaces 7.9 The Orthogonalization Process for Isotropic Symmetric Spaces 7.10 Quadratic Forms Associated with Bilinear Forms 7.11 The Matrix of a Quadratic Form and the Change of Basis 7.12 Diagonalization of a Quadratic Form 7.13 Definiteness of a Real Quadratic Form 8 Sesquilinear and Hermitian Forms 8.1 Sesquilinear Forms and Their Matrices 8.2 The Effect of the Change of Bases 8.3 Orthogonality 9 Tensors and Their Algebras 9.1 The Tensor Product 9.2 Tensor Product of Linear Transformations 9.3 Tensor Algebra 9.4 Exterior Algebra or Grassmann Algebra 10 Applications of Linear Algebra to Numerical Methods 10.1 LU Decomposition of Matrices 10.2 The PLU Decomposition 10.3 Eigenvalue Approximations 10.4 Singular Value Decompositions 10.5 Applications of Singular Value Decomposition 11 Affine and Euclidean Spaces and Applications of Linear Algebra to Geometry 11.1 Affine and Euclidean Spaces 11.2 Affine Transformations 11.3 Isometries 11.4 A Natural Application: Coordinate Transformation in mathbbRmathbbE2 11.5 Affine and Metric Classification of Quadrics 11.6 Projective Classification of Conic Curves and Quadric Surfaces 12 Ordinary Differential Equations and Linear Systems of Ordinary Differential Equations 12.1 A Brief Overview of Basic Concepts of Ordinary Differential Equations 12.2 System of Linear Homogeneous Ordinary Differential Equations 12.3 Real-Valued Solutions for Systems with Complex Eigenvalues 12.4 Homogeneous Differential Equations of nth Order Appendix References Index