Advanced Linear Algebra by Steven Roman

[Greg Bernhardt]

• Author: Steven Roman
• Title: Advanced Linear Algebra
• Amazon link https://www.amazon.com/dp/0387728287/?tag=pfamazon01-20
• Prerequisities: Having completed at least one year of proof based linear algebra. Basic abstract algebra, in particular group and ring theory, is also assumed.
• Level: Grad

Table of Contents:

[LIST]
[*] Preliminaries
[LIST]
[*] Part 1: Preliminaries
[*] Part 2: Algebraic Structures
[/LIST]
[*] Part I: Basic Linear Algebra
[LIST]
[*] Vector Spaces
[LIST]
[*] Vector Spaces
[*] Subspaces
[*] Direct Sums
[*] Spanning Sets and 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
[*] Exercises
[/LIST]
[*] Linear Transformations
[LIST]
[*] Linear Transformations
[*] The Kernel and Image of a Linear Transformation
[*] Isomorphisms
[*] The Rank Plus Nullity Theorem
[*] Linear Transformations from [itex]F^n[/itex] to [itex]F^m[/itex]
[*] 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
[*] Topological Vector Spaces
[*] Linear Operators on [itex]V^\mathbb{C}[/itex]
[*] Exercises
[/LIST]
[*] The Isomorphism Theorems
[LIST]
[*] Quotient Spaces
[*] The Universal Property of Quotients and the First Isomorphism Theorem
[*] Quotient Spaces, Complements and Codimension
[*] Additional Isomorphism Theorems
[*] Linear Functionals
[*] Dual Bases
[*] Reflexivity
[*] Annihilators
[*] Operator Adjoints
[*] Exercises
[/LIST]
[*] Modules I: Basic Properties
[LIST]
[*] Motivation
[*] Modules
[*] Submodules
[*] Spanning Sets
[*] Linear Independence
[*] Torsion Elements
[*] Annihilators
[*] Free Modules
[*] Homomorphisms
[*] Quotient Modules
[*] The Correspondence and Isomorphism Theorems
[*] Direct Sums and Direct Summands
[*] Modules Are Not as Nice as Vector Spaces
[*] Exercises
[/LIST]
[*] Modules II: Free and Noetherian Modules
[LIST]
[*] The Rank of a Free Module
[*] Free Modules and Epimorphisms
[*] Noetherian Modules
[*] The Hilbert Basis Theorem
[*] Exercises
[/LIST]
[*] Modules over a Principal Ideal Domain
[LIST]
[*] Annihilators and Orders
[*] Cyclic Modules
[*] Free Modules over a Principal Ideal Domain
[*] Torsion-Free and Free Modules
[*] The Primary Cyclic Decomposition Theorem
[*] The Invariant Factor Decomposition
[*] Characterizing Cyclic Modules
[*] Indecomposable Modules
[*] Exercises
[/LIST]
[*] The Structure of a Linear Operator
[LIST]
[*] The Module Associated with a Linear Operator
[*] The Primary Cyclic Decomposition of [itex]V_\tau[/itex]
[*] The Characteristic Polynomial
[*] Cyclic and Indecomposable Modules
[*] The Big Picture
[*] The Rational Canonical Form
[*] Exercises
[/LIST]
[*] Eigenvalues and Eigenvectors
[LIST]
[*] Eigenvalues and Eigenvectors
[*] Geometric and Algebraic Multiplicities
[*] The Jordan Canonical Form
[*] Triangularizability and Schur's Theorem
[*] Diagonalizable Operators
[*] Exercises
[/LIST]
[*] Real and Complex Inner Product Spaces
[LIST]
[*] Norm and Distance
[*] Isometries
[*] Orthogonality
[*] Orthogonal and Orthonormal Sets
[*] The Projection Theorem and Best Approximations
[*] The Riesz Representation Theorem
[*] Exercises
[/LIST]
[*] Structure Theory for Normal Operators
[LIST]
[*] The Adjoint of a Linear Operator
[*] Unitary Diagonalizability
[*] Normal Operators
[*] Special Types of Normal Operators
[*] Self-Adjoint Operators
[*] Unitary Operators and Isometries
[*] The Structure of Normal Operators
[*] Functional Calculus
[*] Positive Operators
[*] The Polar Decomposition of an Operator
[*] Exercises
[/LIST]
[/LIST]
[*] Part II: Topics, 257
[LIST]
Metric Vector Spaces: The Theory of Bilinear Forms
[LIST]
[*] Symmetric, Skew-Symmetric and Alternate Forms
[*] The Matrix of a Bilinear Form
[*] Orthogonal Projections
[*] Quadratic Forms
[*] Orthogonality
[*] Linear Functionals
[*] Orthogonal Complements and Orthogonal Direct Sums
[*] Isometries
[*] Hyperbolic Spaces
[*] Nonsingular Completions of a Subspace
[*] The Witt Theorems: A Preview
[*] The Classification Problem for Metric Vector Spaces
[*] Symplectic Geometry
[*] The Structure of Orthogonal Geometries: Orthogonal Bases
[*] The Classification of Orthogonal Geometries: Canonical Forms
[*] The Orthogonal Group
[*] The Witt Theorems for Orthogonal Geometries
[*] Maximal Hyperbolic Subspaces of an Orthogonal Geometry
[*] Exercises
[/LIST]
[*] Metric Spaces
[LIST]
[*] 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
[*] Exercises
[/LIST]
[*] Hilbert Spaces
[LIST]
[*] A Brief Review
[*] Hilbert Spaces
[*] Infinite Series
[*] An Approximation Problem
[*] Hilbert Bases
[*] Fourier Expansions
[*] A Characterization of Hilbert Bases
[*] Hilbert Dimension
[*] A Characterization of Hilbert Spaces
[*] The Riesz Representation Theorem
[*] Exercises
[/LIST]
[*] Tensor Products
[LIST]
[*] Universality
[*] Bilinear Maps
[*] Tensor Products
[*] When Is a Tensor Product Zero?
[*] Coordinate Matrices and Rank
[*] 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
[*] Special Multilinear Maps
[*] Graded Algebras
[*] The Symmetric and Antisymmetric Tensor Algebras
[*] The Determinant
[*] Exercises
[/LIST]
[*] Positive Solutions to Linear Systems: Convexity and Separation
[LIST]
[*] Convex, Closed and Compact Sets
[*] Convex Hulls
[*] Linear and Affine Hyperplanes
[*] Separation
[*] Exercises
[/LIST]
[*] Affine Geometry
[LIST]
[*] Affine Geometry
[*] Affine Combinations
[*] Affine Hulls
[*] The Lattice of Flats
[*] Affine Independence
[*] Affine Transformations
[*] Projective Geometry
[*] Exercises
[/LIST]
[*] Singular Values and the Moore–Penrose Inverse
[LIST]
[*] Singular Values
[*] The Moore–Penrose Generalized Inverse
[*] Least Squares Approximation
[*] Exercises
[/LIST]
[*] An Introduction to Algebras
[LIST]
[*] Motivation
[*] Associative Algebras
[*] Division Algebras
[*] Exercises
[/LIST]
[*] The Umbral Calculus
[LIST]
[*] Formal Power Series
[*] The Umbral Algebra
[*] Formal Power Series as Linear Operators, 477
[*] 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
[*] Umbral Shifts and Derivations of the Umbral Algebra
[*] The Transfer Formulas
[*] A Final Remark
[*] Exercises
[/LIST]
[/LIST]
[*] References
[*] Index of Symbols
[*] Index
[/LIST]

User comments:

• espen180
  This is the most comprehensive and the best written linear algebra book I have seen. The exposition is clear, thorough, and rigorous. It is a great textbook and is also a good reference book.
  
  
• micromass
  This is a very nice book on linear algebra. If you're looking for an advanced text on linear algebra, then this book should be your first choice. As prerequisites, I recommend a rigorous proof-based linear algebra course on the level of Axler or Lang. Further, an abstract algebra course is absolutely required.

 

[espen180]

This should be tagged "Linear Algebra" so that people can find it.

 

[Greg Bernhardt]

This should be tagged "Linear Algebra" so that people can find it.

Good idea, thanks!

 

[QGravity]

A Superb text! this must be taken after reading Hoffman's book. the last chapter on "Umbral Calculus" is a real joy! (the author has a book on it, Umbral Calculus)

 

[rezeew]

The book assumes that the reader is familiar with basic group and ring theory.


I highly recommend this book for anyone looking to deepen their understanding of linear algebra. The author, Steven Roman, is well-known for his clear and concise writing style, making the complex concepts of advanced linear algebra accessible to readers. The book covers a wide range of topics, from basic linear algebra to more advanced topics like tensor products and affine geometry. The prerequisites for this book are quite rigorous, as it assumes a solid understanding of proof-based linear algebra and abstract algebra. However, for those who have completed these courses, this book will serve as an excellent resource and reference guide. Overall, I highly recommend Advanced Linear Algebra by Steven Roman for anyone looking to expand their knowledge and understanding of this important mathematical subject.