TY  - BOOK
AU  - Roman,Steven
TI  - Advanced linear algebra / 
T2  - Graduate texts in mathematics ; 
SN  - 0387247661
U1  - 512.5   22
PY  - 2005///
CY  - New York : 
PB  -  Springer, 
KW  - CIT
KW  - MOT
KW  - ITS
KW  - Algebras, Linear
N1  - 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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