A Guide to Advanced Linear Algebra

A Guide to Advanced Linear Algebra / Steven H. Weintraub.

"This book provides a rigorous and thorough development of linear algebra at an advanced level, and is directed at graduate students and professional mathematicians. It approaches linear algebra from an algebraic point of view, but its selection of topics is governed not only for their importan...

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Bibliographic Details
Main Author: Weintraub, Steven H.
Format: eBook
Language:English
Published: Cambridge : Cambridge University Press, 2012.
Series:Dolciani mathematical expositions.
Subjects:
Algebras, Linear.
MATHEMATICS > Geometry > Algebraic.
MATHEMATICS > Algebra > Linear.
Algebras, Linear
Online Access:Click for online access
Table of Contents:
  • Front cover
  • copyright page
  • title page
  • Preface
  • Contents
  • 1 Vector spaces and linear transformations
  • 1.1 Basic definitions and examples
  • 1.2 Basis and dimension
  • 1.3 Dimension counting and applications
  • 1.4 Subspaces and direct sum decompositions
  • 1.5 Affine subspaces and quotient spaces
  • 1.6 Dual spaces
  • 2 Coordinates
  • 2.1 Coordinates for vectors
  • 2.2 Matrices for linear transformations
  • 2.3 Change of basis
  • 2.4 The matrix of the dual
  • 3 Determinants
  • 3.1 The geometry of volumes
  • 3.2 Existence and uniqueness of determinants3.3 Further properties
  • 3.4 Integrality
  • 3.5 Orientation
  • 3.6 Hilbert matrices
  • 4 The structure of alinear transformation I
  • 4.1 Eigenvalues, eigenvectors, and generalized eigenvectors
  • 4.2 Some structural results
  • 4.3 Diagonalizability
  • 4.4 An application todifferential equations
  • 5 The structure of a linear transformation II
  • 5.1 Annihilating, minimum, and characteristic polynomials
  • 5.2 Invariant subspaces and quotient spaces
  • 5.3 The relationship between the characteristic and minimum polynomials5.4 Invariant subspaces and invariant complements
  • 5.5 Rational canonical form
  • 5.6 Jordan canonical form
  • 5.7 An algorithm for Jordan canonical form and Jordan basis
  • 5.8 Field extensions
  • 5.9 More than one linear transformation
  • 6 Bilinear, sesquilinear, and quadratic forms
  • 6.1 Basic definitions and results
  • 6.2 Characterization and classification theorems
  • 6.3 The adjoint of a linear transformation
  • 7 Real and complex inner product spaces
  • 7.1 Basic definitions
  • 7.2 The Gram-Schmidt process7.3 Adjoints, normal linear transformations, and the spectral theorem
  • 7.4 Examples
  • 7.5 The singular value decomposition
  • 8 Matrix groups as Lie groups
  • 8.1 Definition and first examples
  • 8.2 Isometry groups of forms
  • Appendix A: Polynomials
  • A.1 Basic properties
  • A.2 Unique factorization
  • A.3 Polynomials as expressions and polynomials as functions
  • Appendix B: Modules over principal ideal domains
  • B.1 Definitions and structure theorems
  • B.2 Derivation of canonical forms
  • Bibliography
  • Index

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