A Guide to Advanced Linear Algebra.

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 determinants -- 3.3 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 polynomials -- 5.4 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 process -- 7.3 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 -- About the Author.

Linear algebra occupies a central place in modern mathematics. 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 importance in linear algebra itself, but also for their applications throughout mathematics. Students in algebra, analysis, and topology will find much of interest and use to them, and the careful treatment and breadth of subject matter will make this book a valuable reference for mathematicians throughout their professional lives.Topics treated in this book include: vector spaces and linear transformations; dimension counting and applications; representation of linear transformations by matrices; duality; determinants and their uses; rational and especially Jordan canonical form; bilinear forms; inner product spaces; normal linear transformations and the spectral theorem; and an introduction to matrix groups as Lie groups.The book treats vector spaces in full generality, though it concentrates on the finite dimensional case. Also, it treats vector spaces over arbitrary fields, specializing to algebraically closed fields or to the fields of real and complex numbers as necessary.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.