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Linear Algebra: Algorithms, Applications, and Techniques, Fourth Edition offers a modern and algorithmic approach to computation while providing clear and straightforward theoretical background information. The book guides readers through the major applications, with chapters on properties of real numbers, proof techniques, matrices, vector spaces, linear transformations, eigen values, and Euclidean inner products. Appendices on Jordan canonical forms and Markov chains are included for further study. This useful textbook presents broad and balanced views of theory, with key material highlighted and summarized in each chapter. To further support student practice, the book also includes ample exercises with answers and hints.

• Preface xi Acknowledgments xv
• 1 Logic and Set Theory 1 1.1 Statements 1 Connectives 2 Logical Equivalence 3 1.2 Sets and Quantification 7 Universal Quantification 8 Existential Quantification 9 Negating Quantification 10 Set-Builder Notation 12 Set Operations 13 Families of Sets 14 1.3 Sets and Proofs 18 Direct Proof 20 Subsets 22 Set Equality 23 Indirect Proof 24 Mathematical Induction 25 1.4 Functions 30 Injections 33 Surjections 35 Bijections and Inverses 37 Images and Inverse Images 40 Operations 41
• 2 Euclidean Space 49 2.1 Vectors 49 Vector Operations 51 Distance and Length 57 Lines and Planes 64 2.2 Dot Product 74 Lines and Planes 77 Orthogonal Projection 82 2.3 Cross Product 88 Properties 91 Areas and Volumes 93
• 3 Transformations and Matrices 99 3.1 Linear Transformations 99 Properties 103 Matrices 106 3.2 Matrix Algebra 116 Addition, Subtraction, and Scalar Multiplication 116 Properties 119 Multiplication 122 Identity Matrix 129 Distributive Law 132 Matrices and Polynomials 132 3.3 Linear Operators 137 Re_ections 137 Rotations 142 Isometries 147 Contractions, Dilations, and Shears 150 3.4 Injections and Surjections 155 Kernel 155 Range 158 3.5 Gauss-Jordan Elimination 162 Elementary Row Operations 164 Square Matrices 167 Nonsquare Matrices 171 Gaussian Elimination 177
• 4 Invertibility 183 4.1 Invertible Matrices 183 Elementary Matrices 186 Finding the Inverse of a Matrix 192 Systems of Linear Equations 194 4.2 Determinants 198 Multiplying a Row by a Scalar 203 Adding a Multiple of a Row to Another Row 205 Switching Rows 210 4.3 Inverses and Determinants 215 Uniqueness of the Determinant 216 Equivalents to Invertibility 220 Products 222 4.4 Applications 227 The Classical Adjoint 228 Symmetric and Orthogonal Matrices 229 Cramer's Rule 234 LU Factorization 236 Area and Volume 238
• 5 Abstract Vectors 245 5.1 Vector Spaces 245 Examples of Vector Spaces 247 Linear Transformations 253 5.2 Subspaces 259 Examples of Subspaces 260 Properties 261 Spanning Sets 264 Kernel and Range 266 5.3 Linear Independence 272 Euclidean Examples 274 Abstract Vector Space Examples 276 5.4 Basis and Dimension 281 Basis 281 Zorn's Lemma 285 Dimension 287 Expansions and Reductions 290 5.5 Rank and Nullity 296 Rank-Nullity Theorem 297 Fundamental Subspaces 302 Rank and Nullity of a Matrix 304 5.6 Isomorphism 310 Coordinates 315 Change of Basis 320 Matrix of a Linear Transformation 324
• 6 Inner Product Spaces 335 6.1 Inner Products 335 Norms 341 Metrics 342 Angles 344 Orthogonal Projection 347 6.2 Orthonormal Bases 352 Orthogonal Complement 355 Direct Sum 357 Gram-Schmidt Process 361 QR Factorization 366
• 7 Matrix Theory 373 7.1 Eigenvectors and Eigenvalues 373 Eigenspaces 375 Characteristic Polynomial 377 Cayley-Hamilton Theorem 382 7.2 Minimal Polynomial 386 Invariant Subspaces 389 Generalized Eigenvectors 391 Primary Decomposition Theorem 393 7.3 Similar Matrices 402 Schur's Lemma 405 Block Diagonal Form 408 Nilpotent Matrices 412 Jordan Canonical Form 415 7.4 Diagonalization 422 Orthogonal Diagonalization 426 Simultaneous Diagonalization 428 Quadratic Forms 432 Further Reading 441 Index 443.
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LINEAR ALGEBRA EXPLORE A COMPREHENSIVE INTRODUCTORY TEXT IN LINEAR ALGEBRA WITH COMPELLING SUPPLEMENTARY MATERIALS, INCLUDING A COMPANION WEBSITE AND SOLUTIONS MANUALS Linear Algebra delivers a fulsome exploration of the central concepts in linear algebra, including multidimensional spaces, linear transformations, matrices, matrix algebra, determinants, vector spaces, subspaces, linear independence, basis, inner products, and eigenvectors. While the text provides challenging problems that engage readers in the mathematical theory of linear algebra, it is written in an accessible and simple-to-grasp fashion appropriate for junior undergraduate students. An emphasis on logic, set theory, and functions exists throughout the book, and these topics are introduced early to provide students with a foundation from which to attack the rest of the material in the text. Linear Algebra includes accompanying material in the form of a companion website that features solutions manuals for students and instructors. Finally, the concluding chapter in the book includes discussions of advanced topics like generalized eigenvectors, Schur's Lemma, Jordan canonical form, and quadratic forms. Readers will also benefit from the inclusion of: A thorough introduction to logic and set theory, as well as descriptions of functions and linear transformations An exploration of Euclidean spaces and linear transformations between Euclidean spaces, including vectors, vector algebra, orthogonality, the standard matrix, Gauss-Jordan elimination, inverses, and determinants Discussions of abstract vector spaces, including subspaces, linear independence, dimension, and change of basis A treatment on defining geometries on vector spaces, including the Gram-Schmidt process Perfect for undergraduate students taking their first course in the subject matter, Linear Algebra will also earn a place in the libraries of researchers in computer science or statistics seeking an accessible and practical foundation in linear algebra.
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• Preface
• Computing Generalized Inverses Using Gradient-Based Dynamical Systems
• Cramers Rules for Sylvester-Type Matrix Equations
• BICR Algorithm for Computing Generalized Bisymmetric Solutions of General Coupled Matrix Equations
• System of Mixed Generalized Sylvester-Type Quaternion Matrix Equations
• Hessenberg Matrices: Their Properties and Some Applications
• Equivalence of Polynomial Matrices over a Field
• Matrices in Chemical Problems Modeled Using Directed Graphs and Multigraphs
• Engaging Students in the Learning of Linear Algebra
• Index.
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Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. Systems of linear equations with several unknowns are naturally represented using the formalism of matrices and vectors. So we arrive at the matrix algebra, etc. Linear algebra is central to almost all areas of mathematics. Many ideas and methods of linear algebra were generalised to abstract algebra. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Linear algebra is also used in most sciences and engineering areas because it allows for the modelling of many natural phenomena, and efficiently computes with such models. ''Hot Topics in Linear Algebra'' presents original studies in some areas of the leading edge of linear algebra. Each article has been carefully selected in an attempt to present substantial research results across a broad spectrum. Topics discussed herein include recent advances in analysis of various dynamical systems based on the Gradient Neural Network; Cramer's rules for quaternion generalized Sylvester-type matrix equations by using noncommutative row-column determinants; matrix algorithms for finding the generalized bisymmetric solution pair of general coupled Sylvester-type matrix equations; explicit solution formulas of some systems of mixed generalised Sylvester-type quaternion matrix equations; new approaches to studying the properties of Hessenberg matrices by using triangular tables and their functions; researching of polynomial matrices over a field with respect to semi-scalar equivalence; mathematical modelling problems in chemistry with applying mixing problems, which the associated MP-matrices; and some visual apps, designed in Scilab, for the learning of different topics of linear algebra.
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• Prelude: Linear algebra
• Exterior algebra
• Clifford algebra
• Mappings of inner product spaces
• Spinors in inner product spaces
• Interlude: Analysis
• Exterior calculus
• Hodge decompositions
• Hypercomplex analysis
• Dirac equations
• Multivector calculus on manifolds
• Two index theorems.

This book presents a step-by-step guide to the basic theory of multivectors and spinors, with a focus on conveying to the reader the geometric understanding of these abstract objects. Following in the footsteps of M. Riesz and L. Ahlfors, the book also explains how Clifford algebra offers the ideal tool for studying spacetime isometries and Möbius maps in arbitrary dimensions. The book carefully develops the basic calculus of multivector fields and differential forms, and highlights novelties in the treatment of, e.g., pullbacks and Stokes's theorem as compared to standard literature. It touches on recent research areas in analysis and explains how the function spaces of multivector fields are split into complementary subspaces by the natural first-order differential operators, e.g., Hodge splittings and Hardy splittings. Much of the analysis is done on bounded domains in Euclidean space, with a focus on analysis at the boundary. The book also includes a derivation of new Dirac integral equations for solving Maxwell scattering problems, which hold promise for future numerical applications. The last section presents down-to-earth proofs of index theorems for Dirac operators on compact manifolds, one of the most celebrated achievements of 20th-century mathematics. The book is primarily intended for graduate and PhD students of mathematics. It is also recommended for more advanced undergraduate students, as well as researchers in mathematics interested in an introduction to geometric analysis.

• Vectors and linear systems. Solving linear systems. Vector spaces. Orthogonality. Determinants. Eigenvalues and Eigenvectors. Singular value decomposition.
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Concise Introduction to Linear Algebra deals with the subject of linear algebra, covering vectors and linear systems, vector spaces, orthogonality, determinants, eigenvalues and eigenvectors, singular value decomposition. It adopts an efficient approach to lead students from vectors, matrices quickly into more advanced topics including, LU decomposition, orthogonal decomposition, Least squares solutions, Gram-Schmidt process, eigenvalues and eigenvectors, diagonalizability, spectral decomposition, positive definite matrix, quadratic forms, singular value decompositions and principal component analysis. This book is designed for onesemester teaching to undergraduate students.
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For courses in introductory linear algebra This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price. Please visit www.pearsonhighered.com/math-classics-series for a complete list of titles. Introduction to Linear Algebra, 5th Edition is a foundation book that bridges both practical computation and theoretical principles. Due to its flexible table of contents, the book is accessible for both students majoring in the scientific, engineering, and social sciences, as well as students that want an introduction to mathematical abstraction and logical reasoning. In order to achieve the text's flexibility, the book centers on 3 principal topics: matrix theory and systems of linear equations, elementary vector space concepts, and the eigenvalue problem. This highly adaptable text can be used for a one-quarter or one-semester course at the sophomore/junior level, or for a more advanced class at the junior/senior level.
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• Chapter 1. Vector Spaces.-
• Chapter 2. Linear Transformations.-
• Chapter 3. Elementary Operations.-
• Chapter 4. Inner Product Spaces.-
• Chapter 5. Eigenvalues and Eigenvectors.-
• Chapter 6. Block Diagonal Representation.-
• Chapter 7. Spectral Decomposition.
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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.
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• Matrices and matrix operations.- Determinants.- General vector spaces.- Linear transformations.- Linear transformations and matrices.- Eigenvalues and eigenvectors.- Orthogonal matrices and quadratic forms.
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This self-contained, clearly written textbook on linear algebra is easily accessible for students. It begins with the simple linear equation and generalizes several notions from this equation for the system of linear equations and introduces the main ideas using matrices. It then offers a detailed chapter on determinants and introduces the main ideas with detailed proofs. The third chapter introduces the Euclidean spaces using very simple geometric ideas and discusses various major inequalities and identities. These ideas offer a solid basis for understanding general Hilbert spaces in functional analysis. The following two chapters address general vector spaces, including some rigorous proofs to all the main results, and linear transformation: areas that are ignored or are poorly explained in many textbooks. Chapter 6 introduces the idea of matrices using linear transformation, which is easier to understand than the usual theory of matrices approach. The final two chapters are more advanced, introducing the necessary concepts of eigenvalues and eigenvectors, as well as the theory of symmetric and orthogonal matrices. Each idea presented is followed by examples. The book includes a set of exercises at the end of each chapter, which have been carefully chosen to illustrate the main ideas. Some of them were taken (with some modifications) from recently published papers, and appear in a textbook for the first time. Detailed solutions are provided for every exercise, and these refer to the main theorems in the text when necessary, so students can see the tools used in the solution.
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This book combines a solid theoretical background in linear algebra with practical algorithms for numerical solution of linear algebra problems. Developed from a number of courses taught repeatedly by the authors, the material covers topics like matrix algebra, theory for linear systems of equations, spectral theory, vector and matrix norms combined with main direct and iterative numerical methods, least squares problems, and eigenproblems. Numerical algorithms illustrated by computer programs written in MATLAB (R) are also provided as supplementary material on SpringerLink to give the reader a better understanding of professional numerical software for the solution of real-life problems. Perfect for a one- or two-semester course on numerical linear algebra, matrix computation, and large sparse matrices, this text will interest students at the advanced undergraduate or graduate level.
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• Preface to the Second Edition Preface to the First Edition Acknowledgments List of Figures Symbol Description Vector Spaces Fields The Space n Vector Spaces over an Arbitrary Field Subspaces of Vector Spaces Span and Independence Bases and Finite-Dimensional Vector Spaces Bases and Infinite-Dimensional Vector Spaces Coordinate Vectors Linear Transformations Introduction to Linear Transformations The Range and Kernel of a Linear Transformation The Correspondence and Isomorphism Theorems Matrix of a Linear Transformation The Algebra of L(V, W) and Mmn( ) Invertible Transformations and Matrices Polynomials The Algebra of Polynomials Roots of Polynomials Theory of a Single Linear Operator Invariant Subspaces of an Operator Cyclic Operators Maximal Vectors Indecomposable Linear Operators Invariant Factors and Elementary Divisors Canonical Forms Operators on Real and Complex Vector Spaces Normed and Inner Product Spaces Inner Products Geometry in Inner Product Spaces Orthonormal Sets and the Gram-Schmidt Process Orthogonal Complements and Projections Dual Spaces Adjoints Normed Vector Spaces Linear Operators on Inner Product Spaces Self-Adjoint and Normal Operators Spectral Theorems Normal Operators on Real Inner Product Spaces Unitary and Orthogonal Operators The Polar Decomposition and Singular Value Decomposition Trace and Determinant of a Linear Operator Trace of a Linear Operator Determinant of a Linear Operator and Matrix Uniqueness of the Determinant of a Linear Operator Bilinear Forms Basic Properties of Bilinear Maps Symplectic Spaces Quadratic Forms and Orthogonal Space Orthogonal Space, Characteristic Two Real Quadratic Forms Sesquilinear Forms and Unitary Geometry Basic Properties of Sesquilinear Forms Unitary Space Tensor Products Introduction to Tensor Products Properties of Tensor Products The Tensor Algebra The Symmetric Algebra The Exterior Algebra Clifford Algebras, char <> 2 Linear Groups and Groups of Isometries Linear Groups Symplectic Groups Orthogonal Groups, char <> 2 Unitary Groups Additional Topics in Linear Algebra Matrix Norms The Moore-Penrose Inverse of a Matrix Nonnegative Matrices The Location of Eigenvalues Functions of Matrices Applications of Linear Algebra Least Squares Error Correcting Codes Ranking Webpages for Search Engines Appendices Concepts from Topology and Analysis Concepts from Group Theory Answers to Selected Exercises Hints to Selected Problems Bibliography Index.
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Introducing vector spaces over fields as well as the fundamental concepts of linear combinations, span of vectors, linear independence, basis, and dimension, this book discusses the algebra of polynomials with coefficients in a field, concentrating on results that are consequences of the division algorithm. It develops the structure theory of a linear operator on a finite dimensional vector space and explores inner product spaces from the GramSchmidt process to the spectral theorems for normal and self- adjoint operators.
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Advanced Linear Algebra, Second Edition takes a gentle approach that starts with familiar concepts and then gradually builds to deeper results. Each section begins with an outline of previously introduced concepts and results necessary for mastering the new material. By reviewing what students need to know before moving forward, the text builds a solid foundation upon which to progress. The new edition of this successful text focuses on vector spaces and the maps between them that preserve their structure (linear transformations). Designed for advanced undergraduate and beginning graduate students, the book discusses the structure theory of an operator, various topics on inner product spaces, and the trace and determinant functions of a linear operator. It addresses bilinear forms with a full treatment of symplectic spaces and orthogonal spaces, as well as explains the construction of tensor, symmetric, and exterior algebras. Featuring updates and revisions throughout, Advanced Linear Algebra, Second Edition: * Contains new chapters covering sesquilinear forms, linear groups and groups of isometries, matrices, and three important applications of linear algebra * Adds sections on normed vector spaces, orthogonal spaces over perfect fields of characteristic two, and Clifford algebras * Includes several new exercises and examples, with a solutions manual available upon qualifying course adoption The book shows students the beauty of linear algebra while preparing them for further study in mathematics.
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• -Preface for the Instructor-Preface for the Student-Acknowledgments
• -1. Vector Spaces-
• 2. Finite-Dimensional Vector Spaces-
• 3. Linear Maps-
• 4. Polynomials-
• 5. Eigenvalues, Eigenvectors, and Invariant Subspaces-
• 6. Inner Product Spaces-
• 7. Operators on Inner Product Spaces-
• 8. Operators on Complex Vector Spaces-
• 9. Operators on Real Vector Spaces-
• 10. Trace and Determinant-Photo Credits-Symbol Index-Index.
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This best-selling textbook for a second course in linear algebra is aimed at undergrad math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. The third edition contains major improvements and revisions throughout the book. More than 300 new exercises have been added since the previous edition. Many new examples have been added to illustrate the key ideas of linear algebra. New topics covered in the book include product spaces, quotient spaces, and dual spaces. Beautiful new formatting creates pages with an unusually pleasant appearance in both print and electronic versions. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. Thus the text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator. From reviews of previous editions: "...a didactic masterpiece" -Zentralblatt MATH "...a tour de force in the service of simplicity and clarity ...The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library." -CHOICE "The determinant-free proofs are elegant and intuitive." -American Mathematical Monthly "Clarity through examples is emphasized ...the text is ideal for class exercises ...I congratulate the author and the publisher for a well-produced textbook on linear algebra." -Mathematical Reviews.
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In this appealing and well-written text, Richard Bronson starts with the concrete and computational, and leads the reader to a choice of major applications. The first three chapters address the basics: matrices, vector spaces, and linear transformations. The next three cover eigenvalues, Euclidean inner products, and Jordan canonical forms, offering possibilities that can be tailored to the instructor's taste and to the length of the course. Bronson's approach to computation is modern and algorithmic, and his theory is clean and straightforward. Throughout, the views of the theory presented are broad and balanced and key material is highlighted in the text and summarized at the end of each chapter. The book also includes ample exercises with answers and hints. Prerequisite: One year of calculus is recommended. * Introduces deductive reasoning and helps the reader develop a facility with mathematical proofs* Provides a balanced approach to computation and theory by offering computational algorithms for finding eigenvalues and eigenvectors * Offers excellent exercise sets, ranging from drill to theoretical/challeging along with useful and interesting applications not found in other introductory linear algebra texts.
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Linear algebra permeates mathematics, perhaps more so than any other single subject. It plays an essential role in pure and applied mathematics, statistics, computer science, and many aspects of physics and engineering. This book conveys in a user-friendly way the basic and advanced techniques of linear algebra from the point of view of a working analyst. The techniques are illustrated by a wide sample of applications and examples that are chosen to highlight the tools of the trade. In short, this is material that many of us wish we had been taught as a graduate student. Roughly the first third of the book covers the basic material of a first course in linear algebra. The remaining chapters are devoted to applications drawn from vector calculus, numerical analysis, control theory, complex analysis, convexity and functional analysis. In particular, fixed point theorems, extremal problems, matrix equations, zero location and eigenvalue location problems, and matrices with nonnegative entries are discussed. Appendices on useful facts from analysis and supplementary information from complex function theory are also provided for the convenience of the reader. In this new edition, most of the chapters in the first edition have been revised, some extensively. The revisions include changes in a number of proofs, either to simplify the argument, to make the logic clearer or, on occasion, to sharpen the result. New introductory sections on linear programming, extreme points for polyhedra and a Nevanlinna-Pick interpolation problem have been added, as have some very short introductory sections on the mathematics behind Google, Drazin inverses, band inverses and applications of SVD together with a number of new exercises.
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• Preface
• Chapter 1. Background Linear Algebra
• Chapter 2. The Weyr Form
• Chapter 3. Centralizers
• Chapter 4. The Module Setting
• Chapter 5. Gerstenhaber's Theorem
• Chapter 6. Approximate Simultaneous Diagonalization
• Chapter 7. Algebraic Varieties
• Bibliography.
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Advanced Topics in Linear Algebra presents, in an engaging style, novel topics linked through the Weyr matrix canonical form, a largely unknown cousin of the Jordan canonical form discovered by Eduard Weyr in 1885. The book also develops much linear algebra unconnected to canonical forms, that has not previously appeared in book form. It presents common applications of Weyr form, including matrix commutativity problems, approximate simultaneous diagonalization, and algebraic geometry, with the latter two having topical connections to phylogenetic invariants in biomathematics and multivariate interpolation. The Weyr form clearly outperforms the Jordan form in many situations, particularly where two or more commuting matrices are involved, due to the block upper triangular form a Weyr matrix forces on any commuting matrix. In this book, the authors develop the Weyr form from scratch, and include an algorithm for computing it. The Weyr form is also derived ring-theoretically in an entirely different way to the classical derivation of the Jordan form. A fascinating duality exists between the two forms that allows one to flip back and forth and exploit the combined powers of each. The book weaves together ideas from various mathematical disciplines, demonstrating dramatically the variety and unity of mathematics. Though the book's main focus is linear algebra, it also draws upon ideas from commutative and noncommutative ring theory, module theory, field theory, topology, and algebraic geometry. Advanced Topics in Linear Algebra offers self-contained accounts of the non-trivial results used from outside linear algebra, and lots of worked examples, thereby making it accessible to graduate students. Indeed, the scope of the book makes it an appealing graduate text, either as a reference or for an appropriately designed one or two semester course. A number of the authors' previously unpublished results appear as well.
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• Title Page; Copyright Page; Prefácio; Introdução; Table of Contents; Cálculo numérico e cálculo simbólico; Equação ax = b. Tentemos resolvê-la; Equação ax = b. Atenção aos erros; Equação ax = b. Manipulemos os símbolos; Exercícios; Parte I; 1 Sistemas lineares e matrizes; 1.1 Exemplos de Sistemas Lineares; 1.2 Vetores e matrizes; 1.3 Sistemas lineares genéricos e matrizes associadas; 1.4 Formalismo Ax = b; Exercícios; 2 Operações com matrizes; 2.1 Soma e produto por um número; 2.2 Produto linha por coluna; 2.3 Quanto custa multiplicar duas matrizes?

• 2.4 Algumas propriedades do produto de matrizes2.5 Inversa de uma matriz; Exercícios; 3 Solução dos Sistemas Lineares; 3.1 Matrizes elementares; 3.2 Sistemas lineares quadrados, o método de Gauss; 3.3 Cálculo efetivo da inversa; 3.4 Quanto custa o método de Gauss?; 3.5 Decomposição LU; 3.6 Método de Gauss para sistemas gerais; 3.7 Determinantes; Exercícios; 4 Sistema de coordenadas; 4.1 Escalares e Vetores; 4.2 Coordenadas cartesianas; 4.3 A regra do paralelogramo; 4.4 Sistemas ortogonais, áreas, determinantes; 4.5 Ângulos, módulos, produtos escalares.

• 4.6 Produtos escalares e determinantes em geral4.7 Mudança de coordenadas; 4.8 Espaços vetoriais e bases; Exercícios; Parte II; 5 Formas quadráticas; 5.1 Equações de segundo grau; 5.2 Operações elementares sobre matrizes simétricas; 5.3 Formas quadráticas, funções e positividade; 5.4 Decomposição de Cholesky; Exercícios; 6 Ortogonalidade e ortonormalidade; 6.1 Uplas ortonormais e matrizes ortonormais; 6.2 Rotações; 6.3 Subespaços, independência linear, posto, dimensão; 6.4 Bases ortonormais e Gram-Schmidt; 6.5 Decomposição QR; Exercícios; 7 Projetores, pseudoinversa, mínimos quadrados.

• 7.1 Matrizes e transformações lineares7.2 Projetores; 7.3 Mínimos quadrados e pseudoinversas; Exercícios; 8 Endomorfismos e diagonalização; 8.1 Um exemplo de transformação linear plana; 8.2 Autovalores, autovetores, autoespaços, semelhança; 8.3 Potência de matrizes; 8.4 Os coelhos de Fibonacci; 8.5 Sistemas diferenciais; 8.6 Diagonabilidade das matrizes simétricas reais; Exercícios; Parte III; Apêndice; Conclusão?; Referências; Índice Remissivo.

O objetivo deste livro é o de fornecer as primeiras ferramentas matemáticas associadas à Álgebra Linear. O texto foi elaborado por um matemático que procurou "sair do seu personagem" para vir ao encontro¡ de um público amplo. O desafio é o de tornar acessível a todos, os primeiros fundamentos e as primeiras técnicas de um saber fundamental para a ciência e a tecnologia. Com esse intuito, o autor escolheu escrevê-lo¡ em um estilo não tradicional e o enriqueceu não somente com exercícios, mas também com¡ frases de auto-referência, aforismos, citações, palíndromos¡ e problemas para serem resolvidos com a ajuda do computador. O livro é para todos, mas tem uma dedicação especial aos estudantes dos cursos de graduação em disciplinas científicas e aos professores¡ cujo texto poderá ser útil também como material de consulta