Advanced Linear Algebra, Third Edition

✍ Scribed by Steven Roman

• Publisher: Springer
• Year: 2007
• Tongue: English
• Leaves: 528
• Series: Graduate Texts in Mathematics (GTM), 135
• Edition: 3, 3e, 3rd
• Category: Library

✦ Synopsis

as of December 2023, extensive videos on advanced linear algebra (by the very author!) have also been made available at:
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[url]https://www.youtube.com/@stevenromanmath/playlists[/url]
(alternatively: [url]https://www.youtube.com/channel/UCalphvDmYSEzHchGFW8BxGg[/url])

Coupled with Halmos' Linear Algebra Problem Book (9437D57BCCE39B89E07C889EC57C2786), it is an excellent resource for making an incredibly strong linear algebra basis -- and the mysterious "mathematical maturity" in general. Solve every problem -- or at least try -- and it would pay in a matter of months!

✦ Table of Contents

Cover
Front-matter
Title
Copyright
Author’s Dedication
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Contents
Preliminaries
Part 1. Preliminaries
Multisets
Matrices
Partitioning and Matrix Multiplication
Block Matrices
Elementary Row Operations
Determinants
Polynomials
Functions
Equivalence Relations
Zorn’s Lemma
Cardinality
Cardinal Arithmetic
Part 2. Algebraic Structures
Groups
Cyclic Groups
Rings
Ideals
Quotient Rings and Maximal Ideals
Integral Domains
The Field of Quotients of an Integral Domain
Principal Ideal Domains
Prime and Irreducible Elements
Unique Factorization Domains
Fields
The Characteristic of a Ring
Algebras
Part I: Basic Linear Algebra
Chapter 1. Vector Spaces
Vector Spaces
Examples of Vector Spaces
Subspaces
The Lattice of Subspaces
Direct Sums
External Direct Sums
Internal Direct Sums
Spanning Sets and Linear Independence
Linear Independence
The Dimension of a Vector Space
Ordered Bases and Coordinate Matrices
The Row and Column Spaces of a Matrix
The Complexification of a Real Vector Space
The Dimension of V^C
Exercises
Chapter 2. Linear Transformations
Linear Transformations
The Kernel and Image of a Linear Transformation
Isomorphisms
The Rank Plus Nullity Theorem
Linear Transformations from F^n to F^m
Change of Basis Matrices
The Matrix of a Linear Transformation
Change of Bases for Linear Transformations
Equivalence of Matrices
Similarity of Matrices
Similarity of Operators
Invariant Subspaces and Reducing Pairs
Projection Operators
Projections and Invariance
Orthogonal Projections and Resolutions of the Identity
The Algebra of Projections
Topological Vector Spaces
The Definition
The Standard Topology on R^n
The Natural Topology on V
Linear Operators on V^C
Exercises
Chapter 3. The Isomorphism Theorems
Quotient Spaces
The Natural Projection and the Correspondence Theorem
The Universal Property of Quotients and the First Isomorphism Theorem
Quotient Spaces, Complements and Codimension
Additional Isomorphism Theorems
Linear Functionals
Dual Bases
Reflexivity
Annihilators
Annihilators and Direct Sums
Operator Adjoints
Exercises
Chapter 4. Modules I: Basic Properties
Motivation
Modules
Importance of the Base Ring
Submodules
Spanning Sets
Linear Independence
Torsion Elements
Annihilators
Free Modules
Homomorphisms
Quotient Modules
The Correspondence and Isomorphism Theorems
Direct Sums and Direct Summands
Direct Summands and Extensions of Isomorphisms
Direct Summands and One-Sided Invertibility
Modules Are Not as Nice as Vector Spaces
Exercises
Chapter 5. Modules II: Free and Noetherian Modules
The Rank of a Free Module
Free Modules and Epimorphisms
Noetherian Modules
The Hilbert Basis Theorem
Exercises
Chapter 6. Modules over a Principal Ideal Domain
Annihilators and Orders
Cyclic Modules
The Decomposition of Cyclic Modules
Free Modules over a Principal Ideal Domain
Torsion-Free and Free Modules
The Primary Cyclic Decomposition Theorem
The Primary Decomposition
The Cyclic Decomposition of a Primary Module
The Primary Cyclic Decomposition
Elementary Divisors
The Invariant Factor Decomposition
Characterizing Cyclic Modules
Indecomposable Modules
Indecomposable Submodules of Prime Order
Exercises
More on Complemented Submodules
Chapter 7. The Structure of a Linear Operator
The Module Associated with a Linear Operator
Submodules and Invariant Subspaces
Orders and the Minimal Polynomial
Cyclic Submodules and Cyclic Subspaces
Summary
The Primary Cyclic Decomposition of V_{\tau}
The Characteristic Polynomial
Cyclic and Indecomposable Modules
Indecomposable Modules
Companion Matrices
The Big Picture
The Rational Canonical Form
The Invariant Factor Version
The Determinant Form of the Characteristic Polynomial
Changing the Base Field
Exercises
Chapter 8. Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors
The Trace and the Determinant
Geometric and Algebraic Multiplicities
The Jordan Canonical Form
Triangularizability and Schur’s Lemma
The Real Case
Unitary Triangularizability
Diagonalizable Operators
Spectral Resolutions
Exercises
The Trace of a Matrix
Commuting Operators
Geršgorin Disks
Chapter 9. Real and Complex Inner Product Spaces
Norm and Distance
Isometries
Orthogonality
Orthogonal and Orthonormal Sets
Gram–Schmidt Orthogonalization
The QR Factorization
Hilbert and Hamel Bases
The Projection Theorem and Best Approximations
Characterizing Orthonormal Bases
The Riesz Representation Theorem
Exercises
Extensions of Linear Functionals
Positive Linear Functionals on R^n
Chapter 10. Structure Theory for Normal Operators
The Adjoint of a Linear Operator
The Operator Adjoint and the Hilbert Space Adjoint
Orthogonal Projections
Orthogonal Resolutions of the Identity
Unitary Diagonalizability
Normal Operators
The Spectral Theorem for Normal Operators
The Real Case
Special Types of Normal Operators
Self-Adjoint Operators
Unitary Operators and Isometries
Unitary Similarity
Reflections
The Structure of Normal Operators
Matrix Versions
Functional Calculus
Commutativity
Positive Operators
The Polar Decomposition of an Operator
Exercises
Part II: Topics
Chapter 11. Metric Vector Spaces: The Theory of Bilinear Forms
Symmetric, Skew-Symmetric and Alternate Forms
The Matrix of a Bilinear Form
The Discriminant of a Form
Quadratic Forms
Orthogonality
Orthogonal and Symplectic Geometries
Linear Functionals
Orthogonal Complements and Orthogonal Direct Sums
Isometries
Hyperbolic Spaces
Nonsingular Completions of a Subspace
Extending Isometries to Nonsingular Completions
The Witt Theorems: A Preview
The Classification Problem for Metric Vector Spaces
Symplectic Geometry
The Classification of Symplectic Geometries
Witt’s Extension and Cancellation Theorems
The Structure of the Symplectic Group: Symplectic Transvections
The Structure of Orthogonal Geometries: Orthogonal Bases
Orthogonal Bases
The Classification of Orthogonal Geometries: Canonical Forms
Algebraically Closed Fields
The Real Field R
Finite Fields
The Orthogonal Group
Rotations and Reflections
Symmetries
The Witt Theorems for Orthogonal Geometries
Maximal Hyperbolic Subspaces of an Orthogonal Geometry
Maximal Totally Degenerate Subspaces
Maximal Hyperbolic Subspaces
The Anisotropic Decomposition of an Orthogonal Geometry
Exercises
Chapter 12. Metric Spaces
The Definition
Open and Closed Sets
Convergence in a Metric Space
The Closure of a Set
Dense Subsets
Continuity
Completeness
Isometries
The Completion of a Metric Space
Cauchy Sequences in M
Equivalence Classes of Cauchy Sequences in M
Embedding (M,d) in (M',d')
(M',d') Is Complete
Uniqueness
Exercises
Chapter 13. Hilbert Spaces
A Brief Review
Hilbert Spaces
Infinite Series
An Approximation Problem
Hilbert Bases
Fourier Expansions
The Finite-Dimensional Case
The Countably Infinite-Dimensional Case
The Arbitrary Case
A Characterization of Hilbert Bases
Hilbert Dimension
A Characterization of Hilbert Spaces
The Riesz Representation Theorem
Exercises
Chapter 14. Tensor Products
Universality
Examples of Universality
Bilinear Maps
Tensor Products
Construction I: Intuitive but Not Coordinate Free
Construction II: Coordinate Free
Bilinearity on U \times V Equals Linearity on U \otimes V
When Is a Tensor Product Zero?
Coordinate Matrices and Rank
The Rank of a Decomposable Tensor
Characterizing Vectors in a Tensor Product
Defining Linear Transformations on a Tensor Product
The Tensor Product of Linear Transformations
Change of Base Field
Multilinear Maps and Iterated Tensor Products
Tensor Spaces
Contraction
The Tensor Algebra of V
Special Multilinear Maps
Graded Algebras
The Symmetric and Antisymmetric Tensor Algebras
Symmetric and Antisymmetric Tensors
The Universal Property
The Symmetrization Map
The Determinant
Properties of the Determinant
Exercises
The Tensor Product of Matrices
Chapter 15. Positive Solutions to Linear Systems: Convexity and Separation
Convex, Closed and Compact Sets
Convex Hulls
Linear and Affine Hyperplanes
Separation
Inhomogeneous Systems
Exercises
Chapter 16. Affine Geometry
Affine Geometry
Affine Combinations
Affine Hulls
The Lattice of Flats
Affine Independence
Affine Bases and Barycentric Coordinates
Affine Transformations
Projective Geometry
Exercises
Chapter 17. Singular Values and the Moore–Penrose Inverse
Singular Values
The Moore–Penrose Generalized Inverse
Least Squares Approximation
Exercises
Chapter 18. An Introduction to Algebras
Motivation
Associative Algebras
The Center of an Algebra
From a Vector Space to an Algebra
Examples
The Usual Suspects
Subalgebras
Ideals and Quotients
Homomorphisms
Another View of Algebras
The Regular Representation of an Algebra
Annihilators and Minimal Polynomials
The Spectrum of an Element
Division Algebras
The Quaternions
Finite-Dimensional Division Algebras over an Algebraically Closed Field
Finite-Dimensional Division Algebras over a Finite Field
The Class Equation
The Complex Roots of Unity
Wedderburn’s Theorem
Finite-Dimensional Real Division Algebras
Exercises
Chapter 19. The Umbral Calculus
Formal Power Series
The Umbral Algebra
Formal Power Series as Linear Operators
Sheffer Sequences
Examples of Sheffer Sequences
Umbral Operators and Umbral Shifts
Continuous Operators on the Umbral Algebra
Operator Adjoints
Umbral Operators and Automorphisms of the Umbral Algebra
Sheffer Operators
Umbral Shifts and Derivations of the Umbral Algebra
Sheffer Shifts
The Transfer Formulas
A Final Remark
Exercises
References
General References
General Linear Algebra
Matrix Theory
Multilinear Algebra
Applied and Numerical Linear Algebra
The Umbral Calculus
Index of Symbols
Index