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Linear Algebra Advanced Linear Algebra by Steven Roman

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  1. Jan 18, 2013 #1

    Table of Contents:
    Code (Text):

    [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.
     
    Last edited: Jan 22, 2013
  2. jcsd
  3. Jan 22, 2013 #2
    This should be tagged "Linear Algebra" so that people can find it.
     
  4. Jan 22, 2013 #3
    Good idea, thanks!
     
  5. Feb 8, 2013 #4
    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)
     
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