Advanced linear algebra / Steven Roman.

Includes bibliographical references (p. [473]-474) and index.

Ch. 0. Preliminaries -- Pt. 1. Preliminaries -- Matrices -- Determinants -- Polynomials. Functions -- Equivalence Relations -- Zorn’s Lemma -- Cardinality -- Pt. 2. Algebraic Structures -- Groups -- Rings -- Integral Domains -- Ideals and Principal Ideal Domains -- Prime Elements -- Fields -- The Characteristic of a Ring -- Pt. 1. Basic Linear Algebra -- Ch. 1. Vector Spaces -- Vector Spaces -- Subspaces -- The Lattice of Subspaces -- Direct Sums -- Spanning Sets and Linear Independence -- The Dimension of a Vector Space -- The Row and Column Space of a Matrix -- Coordinate Matrices -- Exercises -- Ch. 2. Linear Transformations -- Linear Transformations -- The Kernel and Image of a Linear Transformation -- Isomorphisms -- The Rank Plus Nullity Theorem -- Linear Transformations from F[superscript n] to F[superscript m] -- Change of Basis Matrices -- The Matrix of a Linear Transformation -- Change of Bases for Linear Transformations -- Equivalence of Matrices -- Similarity of Matrices -- Invariant Subspaces and Reducing Pairs -- Exercises -- Ch. 3. Isomorphism Theorems -- Quotient Spaces -- The First Isomorphism Theorem -- The Dimension of a Quotient Space -- Additional Isomorphism Theorems -- Linear Functionals -- Dual Bases -- Reflexivity -- Annihilators -- Operator Adjoints -- Exercises -- Ch. 4. Modules I -- Motivation -- Modules -- Submodules -- Direct Sums -- Spanning Sets -- Linear Independence -- Homomorphisms -- Free Modules -- Summary -- Exercises -- Ch. 5. Modules II -- Quotient Modules -- Quotient Rings and Maximal Ideals -- Noetherian Modules -- The Hilbert Basis Theorem -- Exercises -- Ch. 6. Modules over Principal Ideal Domains -- Free Modules over a Principal Ideal Domain -- Torsion Modules -- The Primary Decomposition Theorem -- The Cyclic Decomposition Theorem for Primary Modules -- Uniqueness -- The Cyclic Decomposition Theorem -- Exercises -- Ch. 7. Structure of a Linear Operator -- A Brief Review -- The Module Associated with a Linear Operator -- Submodules and Invariant Subspaces -- Orders and the Minimal Polynomial -- Cyclic Submodules and Cyclic Subspaces -- Summary -- The Decomposition of V. -- The Rational Canonical Form -- Exercises -- Ch. 8. Eigenvalues and Eigenvectors -- The Characteristic Polynomial of an Operator -- Eigenvalues and Eigenvectors -- The Cayley-Hamilton Theorem -- The Jordan Canonical Form -- Geometric and Algebraic Multiplicities -- Diagonalizable Operators -- Projections -- The Algebra of Projections -- Resolutions of the Identity -- Projections and Diagonalizability -- Projections and Invariance -- Exercises -- Ch. 9. Real and Complex Inner Product Spaces -- Introduction -- Norm and Distance -- Isometries -- Orthogonality -- Orthogonal and Orthonormal Sets -- The Projection Theorem -- The Gram-Schmidt Orthogonalization Process -- The Riesz Representation Theorem -- Exercises -- Ch. 10. Spectral Theorem for Normal Operators -- The Adjoint of a Linear Operator -- Orthogonal Diagonalizability -- Motivation -- Self-Adjoint Operators -- Unitary Operators -- Normal Operators -- Orthogonal Diagonalization -- Orthogonal Projections -- Orthogonal Resolutions of the Identity -- The Spectral Theorem -- Functional Calculus -- Positive Operators -- The Polar Decomposition of an Operator -- Exercises -- Pt. 2. Topics -- Ch. 11. Metric Vector Spaces -- Symmetric, Skew-symmetric and Alternate Forms -- The Matrix of a Bilinear Form -- Quadratic Forms -- Linear Functionals -- Orthogonality -- Orthogonal Complements -- Orthogonal Direct Sums -- Quotient Spaces -- Symplectic Geometry - Hyperbolic Planes -- Orthogonal Geometry - Orthogonal Bases -- The Structure of an Orthogonal Geometry -- Isometries -- Symmetries -- Witt’s Cancellation Theorem -- Witt’s Extension Theorem -- Maximum Hyperbolic Subspaces -- Exercises -- Ch. 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 -- Exercises -- Ch. 13. Hilbert Spaces -- 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 -- Ch. 14. Tensor Products -- Free Vector Spaces -- Another Look at the Direct Sum -- Bilinear Maps and Tensor Products -- Properties of the Tensor Product -- The Tensor Product of Linear Transformations -- Change of Base Field -- Multilinear Maps and Iterated Tensor Products -- Alternating Maps and Exterior Products -- Exercises -- Ch. 15. Affine Geometry -- Affine Geometry -- Affine Combinations -- Affine Hulls -- The Lattice of Flats -- Affine Independence -- Affine Transformations -- Projective Geometry -- Exercises -- Ch. 16. 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 -- Automorphisms of the Umbral Algebra -- Derivations of the Umbral Algebra -- Exercises.

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