Advanced Linear Algebra, Third Edition

as of December 2023, extensive videos on advanced linear algebra (by the very author!) have also been made available at: [url]https://www.youtube.com/watch?list=PLiyVurqwtq0ZPlwaAojwrWLjPP9ZKlJ\_N[/url] the author's YouTube channel videos are available at: [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! 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 This is a graduate textbook covering an especially broad range of topics. The first part of the book contains a careful but rapid discussion of the basics of linear algebra, including vector spaces, linear transformations, quotient spaces, and isomorphism theorems. The author then proceeds to modules, emphasizing a comparison with vector spaces. A thorough discussion of inner product spaces, eigenvalues, eigenvectors, and finite dimensional spectral theory follows, culminating in the finite dimensional spectral theorem for normal operators. The second part of the book is a collection of topics, including metric vector spaces, metric spaces, Hilbert spaces, tensor products, and affine geometry. The last chapter discusses the umbral calculus, an area of modern algebra with important applications. For the third edition, the author has added a new chapter on associative algebras that includes the well known characterizations of the finite-dimensional division algebras over the real field (a theorem of Frobenius) and over a finite field (Wedderburn's theorem); polished and refined some arguments (such as the discussion of reflexivity, the rational canonical form, best approximations and the definitions of tensor products); upgraded some proofs that were originally done only for finite-dimensional/rank cases; added new theorems, including the spectral mapping theorem; and considerably expanded the reference section with over a hundred references to books on linear algebra. From the reviews of the second edition: "In this 2nd edition, the author has rewritten the entire book and has added more than 100 pages of new materials. ... As in the previous edition, the text is well written and gives a thorough discussion of many topics of linear algebra and related fields. ... the exercises are rewritten and expanded. ... Overall, I found the book a very useful one. ... It is a suitable choice as a graduate text or as a reference book." - Ali-Akbar Jafarian, ZentralblattMATH "This is a formidable volume, a compendium of linear algebra theory, classical and modern ... . The development of the subject is elegant ... . The proofs are neat ... . The exercise sets are good, with occasional hints given for the solution of trickier problems. ... It represents linear algebra and does so comprehensively." -Henry Ricardo, MathDL For the third edition, the author has added a new chapter on associative algebras that includes the well known characterizations of the finite-dimensional division algebras over the real field (a theorem of Frobenius) and over a finite field (Wedderburn's theorem); polished and refined some arguments (such as the discussion of reflexivity, the rational canonical form, best approximations and the definitions of tensor products); upgraded some proofs that were originally done only for finite-dimensional/rank cases; added new theorems, including the spectral mapping theorem; corrected all known errors; the reference section has been enlarged considerably, with over a hundred references to books on linear algebra. From the reviews of the second edition: “In this 2nd edition, the author has rewritten the entire book and has added more than 100 pages of new materials. ... As in the previous edition, the text is well written and gives a thorough discussion of many topics of linear algebra and related fields. ... the exercises are rewritten and expanded. ... Overall, I found the book a very useful one. ... It is a suitable choice as a graduate text or as a reference book.” Ali-Akbar Jafarian, ZentralblattMATH “This is a formidable volume, a compendium of linear algebra theory, classical and modern .... The development of the subject is elegant .... The proofs are neat .... The exercise sets are good, with occasional hints given for the solution of trickier problems. ... It represents linear algebra and does so comprehensively.” Henry Ricardo, MathDL

For the third edition, the author has added a new chapter on associative algebras that includes the well known characterizations of the finite-dimensional division algebras over the real field (a theorem of Frobenius) and over a finite field (Wedderburn's theorem); polished and refined some arguments (such as the discussion of reflexivity, the rational canonical form, best approximations and the definitions of tensor products); upgraded some proofs that were originally done only for finite-dimensional/rank cases; added new theorems, including the spectral mapping theorem; considerably expanded the reference section with over a hundred references to books on linear algebra.

From the reviews of the second edition:

"In this 2nd edition, the author has rewritten the entire book and has added more than 100 pages of new materials....As in the previous edition, the text is well written and gives a thorough discussion of many topics of linear algebra and related fields...the exercises are rewritten and expanded....Overall, I found the book a very useful one....It is a suitable choice as a graduate text or as a reference book."

Ali-Akbar Jafarian, ZentralblattMATH

"This is a formidable volume, a compendium of linear algebra theory, classical and modern... The development of the subject is elegant...The proofs are neat...The exercise sets are good, with occasional hints given for the solution of trickier problems...It represents linear algebra and does so comprehensively."

Henry Ricardo, MAA Online

This Graduate Level Textbook Covers An Especially Broad Range Of Topics. The Book First Offers A Careful Discussion Of The Basics Of Linear Algebra. It Then Proceeds To A Discussion Of Modules, Emphasizing A Comparison With Vector Spaces, And Presents A Thorough Discussion Of Inner Product Spaces, Eigenvalues, Eigenvectors, And Finite Dimensional Spectral Theory, Culminating In The Finite Dimensional Spectral Theorem For Normal Operators. The New Edition Has Been Revised And Contains A Chapter On The Qr Decomposition, Singular Values And Pseudoinverses, And A Chapter On Convexity, Separation And Positive Solutions To Linear Systems.