Advanced Linear Algebra

Designed for advanced undergraduate and beginning graduate students in linear or abstract algebra, Advanced Linear Algebra covers theoretical aspects of the subject, along with examples, computations, and proofs. It explores a variety of advanced topics in linear algebra that highlight the rich interconnections of the subject to geometry, algebra, analysis, combinatorics, numerical computation, and many other areas of mathematics. The book’s 20 chapters are grouped into six main areas: algebraic structures, matrices, structured matrices, geometric aspects of linear algebra, modules, and multilinear algebra. The level of abstraction gradually increases as students proceed through the text, moving from matrices to vector spaces to modules. Each chapter consists of a mathematical vignette devoted to the development of one specific topic. Some chapters look at introductory material from a sophisticated or abstract viewpoint while others provide elementary expositions of more theoretical concepts. Several chapters offer unusual perspectives or novel treatments of standard results. Unlike similar advanced mathematical texts, this one minimizes the dependence of each chapter on material found in previous chapters so that students may immediately turn to the relevant chapter without first wading through pages of earlier material to access the necessary algebraic background and theorems. Chapter summaries contain a structured list of the principal definitions and results. End-of-chapter exercises aid students in digesting the material. Students are encouraged to use a computer algebra system to help solve computationally intensive exercises.

Background on Algebraic Structures Overview of Algebraic Systems Groups Rings and Fields Vector Spaces Subsystems Product Systems Quotient Systems Homomorphisms Spanning, Linear Independence, Basis, and Dimension Permutations Symmetric Groups Representing Functions as Directed Graphs Cycle Decompositions of Permutations Composition of Cycles Factorizations of Permutations Inversions and Sorting Signs of Permutations Polynomials Intuitive Definition of Polynomials Algebraic Operations on Polynomials Formal Power Series and Polynomials Properties of Degree Evaluating Polynomials Polynomial Division with Remainder Divisibility and Associates Greatest Common Divisors of Polynomials GCDs of Lists of Polynomials Matrix Reduction Algorithm for GCDs Roots of Polynomials Irreducible Polynomials Factorization of Polynomials into Irreducibles Prime Factorizations and Divisibility Irreducible Polynomials in Q[x] Irreducibility in Q[x] via Reduction Mod p Eisenstein’s Irreducibility Criterion for Q[x] Kronecker’s Algorithm for Factoring in Q[x] Algebraic Elements and Minimal Polynomials Multivariable Polynomials Matrices Basic Matrix Operations Formal Definition of Matrices and Vectors Vector Spaces of Functions Matrix Operations via Entries Properties of Matrix Multiplication Generalized Associativity Invertible Matrices Matrix Operations via Columns Matrix Operations via Rows Elementary Operations and Elementary Matrices Elementary Matrices and Gaussian Elimination Elementary Matrices and Invertibility Row Rank and Column Rank Conditions for Invertibility of a Matrix Determinants via Calculations Matrices with Entries in a Ring Explicit Definition of the Determinant Diagonal and Triangular Matrices Changing Variables Transposes and Determinants Multilinearity and the Alternating Property Elementary Row Operations and Determinants Determinant Properties Involving Columns Product Formula via Elementary Matrices Laplace Expansions Classical Adjoints and Inverses Cramer’s Rule Product Formula for Determinants Cauchy–Binet Formula Cayley–Hamilton Theorem Permanents Concrete vs. Abstract Linear Algebra Concrete Column Vectors vs. Abstract Vectors Examples of Computing Coordinates Concrete vs. Abstract Vector Space Operations Matrices vs. Linear Maps Examples of Matrices Associated with Linear Maps Vector Operations on Matrices and Linear Maps Matrix Transpose vs. Dual Maps Matrix/Vector Multiplication vs. Evaluation of Maps Matrix Multiplication vs. Composition of Linear Maps Transition Matrices and Changing Coordinates Changing Bases Algebras of Matrices and Linear Operators Similarity of Matrices and Linear Maps Diagonalizability and Triangulability Block-Triangular Matrices and Invariant Subspaces Block-Diagonal Matrices and Reducing Subspaces Idempotent Matrices and Projections Bilinear Maps and Matrices Congruence of Matrices Real Inner Product Spaces and Orthogonal Matrices Complex Inner Product Spaces and Unitary Matrices Matrices with Special Structure Hermitian, Positive Definite, Unitary, and Normal Matrices Conjugate-Transpose of a Matrix Hermitian Matrices Hermitian Decomposition of a Matrix Positive Definite Matrices Unitary Matrices Unitary Similarity Unitary Triangularization Simultaneous Triangularization Normal Matrices and Unitary Diagonalization Polynomials and Commuting Matrices Simultaneous Unitary Diagonalization Polar Decomposition: Invertible Case Polar Decomposition: General Case Interlacing Eigenvalues for Hermitian Matrices Determinant Criterion for Positive Definite Matrices Jordan Canonical Forms Examples of Nilpotent Maps Partition Diagrams Partition Diagrams and Nilpotent Maps Computing Images via Partition Diagrams Computing Null Spaces via Partition Diagrams Classification of Nilpotent Maps (Stage 1) Classification of Nilpotent Maps (Stage 2) Classification of Nilpotent Maps (Stage 3) Fitting’s Lemma Existence of Jordan Canonical Forms Uniqueness of Jordan Canonical Forms Computing Jordan Canonical Forms Application to Differential Equations Minimal Polynomials Jordan–Chevalley Decomposition of a Linear Operator Matrix Factorizations Approximation by Orthonormal Vectors Gram–Schmidt Orthonormalization Gram–Schmidt QR Factorization Householder Reflections Householder QR Factorization LU Factorization Example of the LU Factorization LU Factorizations and Gaussian Elimination Permuted LU Factorizations Cholesky Factorization Least Squares Approximation Singular Value Decomposition Iterative Algorithms in Numerical Linear Algebra Richardson’s Algorithm Jacobi’s Algorithm Gauss–Seidel Algorithm Vector Norms Metric Spaces Convergence of Sequences Comparable Norms Matrix Norms Formulas for Matrix Norms Matrix Inversion via Geometric Series Affine Iteration and Richardson’s Algorithm Splitting Matrices and Jacobi’s Algorithm Induced Matrix Norms and the Spectral Radius Analysis of the Gauss–Seidel Algorithm Power Method for Finding Eigenvalues Shifted and Inverse Power Method Deflation The Interplay of Geometry and Linear Algebra Affine Geometry and Convexity Linear Subspaces Examples of Linear Subspaces Characterizations of Linear Subspaces Affine Combinations and Affine Sets Affine Sets and Linear Subspaces Affine Span of a Set Affine Independence Affine Bases and Barycentric Coordinates Characterizations of Affine Sets Affine Maps Convex Sets Convex Hulls Carath´eodory’s Theorem Hyperplanes and Half-Spaces in Rn Closed Convex Sets Cones and Convex Cones Intersection Lemma for V-Cones All H-Cones Are V-Cones Projection Lemma for H-Cones All V-Cones Are H-Cones Finite Intersections of Closed Half-Spaces Convex Functions Derivative Tests for Convex Functions Ruler and Compass Constructions Geometric Constructibility Arithmetic Constructibility Preliminaries on Field Extensions Field-Theoretic Constructibility Proof that GC ? AC Proof that AC ? GC Algebraic Elements and Minimal Polynomials Proof that AC = SQC Impossibility of Geometric Construction Problems Constructibility of the 17-Gon Overview of Solvability by Radicals Dual Spaces and Bilinear Forms Vector Spaces of Linear Maps Dual Bases Zero Sets Annihilators Double Dual V ** Correspondence between Subspaces of V and V * Dual Maps Nondegenerate Bilinear Forms Real Inner Product Spaces Complex Inner Product Spaces Comments on Infinite-Dimensional Spaces Affine Algebraic Geometry Metric Spaces and Hilbert Spaces Metric Spaces Convergent Sequences Closed Sets Open Sets Continuous Functions Compact Sets Completeness Definition of a Hilbert Space Examples of Hilbert Spaces Proof of the Hilbert Space Axioms for l2(X) Basic Properties of Hilbert Spaces Closed Convex Sets in Hilbert Spaces Orthogonal Complements Orthonormal Sets Maximal Orthonormal Sets Isomorphism of H and l2(X) Continuous Linear Maps Dual Space of a Hilbert Space Adjoints Modules, Independence, and Classification Theorems Finitely Generated Commutative Groups Commutative Groups Generating Sets Z-Independence and Z-Bases Elementary Operations on Z-Bases Coordinates and Z-Linear Maps UMP for Free Commutative Groups Quotient Groups of Free Commutative Groups Subgroups of Free Commutative Groups Z-Linear Maps and Integer Matrices Elementary Operations and Change of Basis Reduction Theorem for Integer Matrices Structure of Z-Linear Maps between Free Groups Structure of Finitely Generated Commutative Groups Example of the Reduction Algorithm Some Special Subgroups Uniqueness Proof: Free Case Uniqueness Proof: Prime Power Case Uniqueness of Elementary Divisors Uniqueness of Invariant Factors Uniqueness Proof: General Case Axiomatic Approach to Independence, Bases, and Dimension Axioms Definitions Initial Theorems Consequences of the Exchange Axiom Main Theorems: Finite-Dimensional Case Zorn’s Lemma Main Theorems: General Case Bases of Subspaces Linear Independence and Linear Bases Field Extensions Algebraic Independence and Transcendence Bases Independence in Graphs Hereditary Systems Matroids Equivalence of Matroid Axioms Elements of Module Theory Module Axioms Examples of Modules Submodules Submodule Generated by a Subset Direct Products, Direct Sums, and Hom Modules Quotient Modules Changing the Ring of Scalars Fundamental Homomorphism Theorem for Modules More Module Homomorphism Theorems Chains of Submodules Modules of Finite Length Free Modules Size of a Basis of a Free Module Principal Ideal Domains, Modules over PIDs, and Canonical Forms Principal Ideal Domains Divisibility in Commutative Rings Divisibility and Ideals Prime and Irreducible Elements Irreducible Factorizations in PIDs Free Modules over a PID Operations on Bases Matrices of Linear Maps between Free Modules Reduction Theorem for Matrices over a PID Structure Theorems for Linear Maps and Modules Minors and Matrix Invariants Uniqueness of Smith Normal Form Torsion Submodules Uniqueness of Invariant Factors Uniqueness of Elementary Divisors F[x]-Module Defined by a Linear Operator Rational Canonical Form of a Linear Map Jordan Canonical Form of a Linear Map Canonical Forms of Matrices Universal Mapping Properties and Multilinear Algebra Introduction to Universal Mapping Properties Bases of Free R-Modules Homomorphisms out of Quotient Modules Direct Product of Two Modules Direct Sum of Two Modules Direct Products of Arbitrary Families of R-Modules Direct Sums of Arbitrary Families of R-Modules Solving Universal Mapping Problems Universal Mapping Problems in Multilinear Algebra Multilinear Maps Alternating Maps Symmetric Maps Tensor Product of Modules Exterior Powers of a Module Symmetric Powers of a Module Myths about Tensor Products Tensor Product Isomorphisms Associativity of Tensor Products Tensor Product of Maps Bases and Multilinear Maps Bases for Tensor Products of Free R-Modules Bases and Alternating Maps Bases for Exterior Powers of Free Modules Bases for Symmetric Powers of Free Modules Tensor Product of Matrices Determinants and Exterior Powers From Modules to Algebras Appendix: Basic Definitions Further Reading Bibliography Index Summary and Exercises appear at the end of each chapter.