The Oxford Linear Algebra for Scientists

## Abstract This book provides a introduction into linear algebra which covers the mathematical set-up as well as applications to science. After the introductory material on sets, functions, groups and fields, the basic features of vector spaces are developed, including linear independence, bases, dimension, vector subspaces and linear maps. Practical methods for calculating with dot, cross and triple products are introduced early on. The theory of linear maps and their relation to matrices is developed in detail, culminating in the rank theorem. Algorithmic methods bases on row reduction and determinants are discussed an applied to computing the rank and the inverse of matrices and to solve systems of linear equations. Eigenvalues and eigenvectors and the application to diagonalising linear maps, as well as scalar products and unitary linear maps are covered in detail. Advanced topics included are the Jordon normal form, normal linear maps, the singular value decomposition, bi-linear and sesqui-linear forms, duality and tensors. The book also included short expositions of diverse scientific applications of linear algebra, including to internet search, classical mechanics, graph theory, cryptography, coding theory, data compression, special relativity, quantum mechanics and quantum computing. cover titlepage copyright dedication preface Acknowledgements contents Linearity — an informal introduction Why linearity? Linearity, more abstractly Linear functions Linear equations Vectors with two components Linearity for maps between vectors Linear maps and matrices Back to linear equations Plan of the book Exercises Part I Preliminaries Sets and functions Sets (Non-) definition of sets Set operations New sets from old ones Relations Basic definitions Properties of equivalence relations Functions Definition of functions Composition of functions Properties of functions The inverse function Rudiments of logic Predicates and Boolean operations Implications Quantifiers Patterns of proofs Exercises Groups Definition and basic properties Definition Examples of groups Sub-groups Group homomorphisms Permutation groups Calculating with permutations Permutations in terms of transpositions The sign of permutations Exercises Fields Fields and their properties Definition Some conclusions from the field axioms Order on fields Examples of fields The complex numbers Construction of complex numbers Complex conjugation Beyond R2 Basics of polynomials Basics and polynomial division Zeros and multiplicity Factorization Exercises Part II Vector spaces Coordinate vectors Basic definitions Definition of coordinate vectors Addition and scalar multiplication Calculating with coordinate vectors Standard unit vectors Definition of standard unit vectors Calculating with standard unit vectors Exercises Vector spaces Basic definitions Vector space axioms Implications of vector space axioms Vector subspaces Linear Maps Algebras Examples of vector spaces Coordinate vector spaces Matrices and matrix vector spaces Vector spaces of functions Exercises Elementary vector space properties Linear independence Linear combinations and span Linear independence Properties of linearly independent vectors Examples for linear independence Basis and dimension Basis and coordinates Examples of bases and coordinates Dimension of a vector space Existence of a basis Properties of finite-dimensional vector spaces Exercises Vector subspaces Intersection and sum Intersection of vector subspaces Union and sum Dimension of vector space sums Direct sums Direct sums of vector spaces Quotient spaces* Equivalence relation and cosets Quotient vector space Exercises Part III Basic geometry The dot product Basic properties Definition of dot product Properties of the dot product Length and angle Definition of length The Cauchy–Schwarz inequality Properties of the length The angle between vectors Orthogonality Definition of orthogonality The Kronecker delta symbol Orthonormal basis Exercises Vector and triple product The cross product Orthogonality in two dimensions Definition of cross product in R3 Existence and uniqueness of the cross product The Levi-Civita symbol in R3 Properties of the cross product Geometrical interpretation of the cross product The triple product Definition of triple product Calculation of the determinant Interpretation of the triple product Exercises Lines and planes Lines in R2 Parametric and Cartesian form Intersection of two lines Lines and planes in R3 Parametric and Cartesian form for planes Parametric and Cartesian form for lines Minimal distances Intersection of two planes Intersection of line and plane Intersection of three planes Exercises Part IV Linear maps and matrices Introduction to linear maps First properties of linear maps Reminder of definition Existence and construction of linear maps Addition and scalar multiplication of linear maps Map composition and inverse Isomorphisms and general linear groups Examples of linear maps Coordinate maps Differential operators Exercises Matrices Matrices as linear maps Linear maps between coordinate vectors Matrix-vector multiplication The two faces of matrices Square and diagonal matrices Matrix multiplication Matrix multiplication from map composition Rules for matrix multiplication Matrix inverse and general linear group Transpose and Hermitian conjugate The transpose of a matrix Symmetric and anti-symmetric matrices Properties of transposition The Hermitian conjugate of a matrix Hermitian and anti-Hermitian matrices Properties of Hermitian conjugation Exercises The structure of linear maps Image and kernel Definition of image and kernel Rank of a linear map Injective and surjective linear maps The rank theorem Motivation The theorem Easy conclusions from the rank theorem Isomorphisms The inverse of a linear map Another proof of the rank theorem* Exercises Linear maps in terms of matrices Matrices representing linear maps Basis choice Computing the representing matrix Examples for matrices describing linear maps Change of basis General case Identical domain and co-domain Conjugate matrices Exercises Part V Linear systems and algorithms Computing with matrices Row operations Definition of row operations Upper echelon form Algorithm to bring a matrix into upper echelon form Rank of a matrix Row and column rank Computing the rank Matrix inverse The elementary matrices Algorithm to calculate the matrix inverse Exercises Linear systems Abstract linear systems Definition of linear systems Structure of solution space Systems of linear equations Definition Solutions of homogeneous system Solution of inhomogeneous system Examples with explicit calculation Row operations for linear equations Algorithm for solving linear equations Applications to geometry Parametric and Cartesian form Intersection of affine k-planes Intersections and linear systems Exercises Determinants Existence and uniqueness Definition of determinant The general formula for the determinant The Levi-Civita symbol The determinant in low dimensions Determinants for triangular matrices Properties of the determinant Determinant and transposition Determinant and matrix multiplication Determinant and basis transformation Orientation Computing with determinants The co-factor matrix Laplace expansion of determinant Matrix inverse from co-factors Determinant and row operations Minors Cramer's rule Exercises Part VI Eigenvalues and eigenvectors Basics of eigenvalues Eigenvalues and eigenspaces Definition of eigenvalues and eigenvectors Degeneracy and eigenspaces The characteristic polynomial Definition of characteristic polynomial Properties of the characteristic polynomial Examples Degeneracy and multiplicity Class functions The theorem of Cayley–Hamilton* Polyomials of endomorphisms The minimal polynomial The theorem Exercises Diagonalizing linear maps Diagonalization Basic criteria The diagonal matrix Diagonalizing and class functions Examples Projectors Definition of projectors Diagonalizing projectors Simultaneous diagonalization* Diagonalization of restricted maps Criterion for simultaneous diagonalization Exercises The Jordan normal form* Nilpotent endormorphisms* Powers of endomorphisms Definition of nilpotentency Structure of nilpotent endomorphisms Examples The Jordan form* The decomposition theorem The theorem Implications of Jordan normal form Examples* Exercises Part VII Inner product vector spaces Scalar products Real and Hermitian scalar products Definition of norms Definition of scalar products The norm associated to a scalar product Orthogonal vectors and angles Examples of scalar products Orthogonality and Gram–Schmidt procedure Ortho-normal bases Existence of ortho-normal bases Construction of ortho-normal bases Properties of ortho-normal bases Orthogonal spaces Exercises Adjoint and unitary maps Adjoint and self-adjoint maps Definition and basic properties of adjoint map Adjoint map relative to a basis Examples Kernel and image of the adjoint map Self-adjoint maps Unitary maps Definition of unitary maps Unitary groups Orthogonal matrices Unitary matrices Exercises Diagonalization — again Hermitian maps Eigenvectors and eigenvalues of Hermitian maps Diagonalizing Hermitian maps Examples Normal maps* Definition of normal maps Diagonalization of normal maps Diagonalizing unitary maps Orthogonal matrices Three-dimensional rotations — again Singular value decomposition* General bases Ortho-normal bases Functions of matrices* Defining functions of matrices Matrix functions and diagonalization Direct computation of matrix functions Exercises Bi-linear and sesqui-linear forms* Basics definitions* Definition of bi-linear and sesqui-linear forms The associated quadratic form Linear forms relative to a basis Positive definiteness Degeneracy Classification of linear forms* A normal form for the describing matrix Theorem of Sylvester Groups associated to linear forms Quadratic hyper-surfaces* Definition of quadratic hyper-surfaces Diagonalization of quadratic hyper-surfaces Quadratic curves in R2 Quadratic surfaces in R3 Exercises Part VIII Dual and tensor vector spaces* The dual vector space* Definition of dual vector space* Linear functionals Dual basis Index notation The double dual The dual map* The orthogonal space The dual map Kernel and image of the dual map Linear forms and dual space* The map between V and V* Index notation — again Exercises Tensors* Tensor basics* Definition of tensors The tensor product The universal property Indices Basis transformation of tensors Induced maps on tensors Further tensor properties* Symmetric and anti-symmetric tensors Linear maps as tensors Bi-linear forms and tensors Multi-linearity* Higher-rank tensors (p,q) tensors Alternating q forms The determinant as an alternating form The outer algebra of R3 Exercises References Index "This book provides a introduction into linear algebra which covers the mathematical set-up as well as applications to science. After the introductory material on sets, functions, groups and fields, the basic features of vector spaces are developed, including linear independence, bases, dimension, vector subspaces and linear maps. Practical methods for calculating with dot, cross and triple products are introduced early on. The theory of linear maps and their relation to matrices is developed in detail, culminating in the rank theorem. Algorithmic methods bases on row reduction and determinants are discussed an applied to computing the rank and the inverse of matrices and to solve systems of linear equations. Eigenvalues and eigenvectors and the application to diagonalising linear maps, as well as scalar products and unitary linear maps are covered in detail. Advanced topics included are the Jordon normal form, normal linear maps, the singular value decomposition, bi-linear and sesqui-linear forms, duality and tensors. The book also included short expositions of diverse scientific applications of linear algebra, including to internet search, classical mechanics, graph theory, cryptography, coding theory, data compression, special relativity, quantum mechanics and quantum computing"-- Provided by publisher « This textbook provides a modern introduction to linear algebra, a mathematical discipline every first year undergraduate student in physics and engineering must learn. A rigorous introduction into the mathematics is combined with many examples, solved problems, and exercises as well as scientific applications of linear algebra. These include applications to contemporary topics such as internet search, artificial intelligence, neural networks, and quantum computing, as well as a number of more advanced topics, such as Jordan normal form, singular value decomposition, and tensors, which will make it a useful reference for a more experienced practitioner. Structured into 27 chapters, it is designed as a basis for a lecture course and combines a rigorous mathematical development of the subject with a range of concisely presented scientific applications. The main text contains many examples and solved problems to help the reader develop a working knowledge of the subject and every chapter comes with exercises. »-- Amazon