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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