1st Edition

Advanced Linear Algebra

By Nicholas A. Loehr Copyright 2014
    632 Pages 25 B/W Illustrations
    by Chapman & Hall

    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
    Rings and Fields
    Vector Spaces
    Product Systems
    Quotient Systems
    Spanning, Linear Independence, Basis, and Dimension

    Symmetric Groups
    Representing Functions as Directed Graphs
    Cycle Decompositions of Permutations
    Composition of Cycles
    Factorizations of Permutations
    Inversions and Sorting
    Signs of Permutations

    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

    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

    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

    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
    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
    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

    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
    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
    Equivalence of Matroid Axioms

    Elements of Module Theory
    Module Axioms
    Examples of Modules
    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



    Summary and Exercises appear at the end of each chapter.


    Nicholas Loehr