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.
Biography
Nicholas Loehr
