UID:
almahu_9949616130902882
Umfang:
1 online resource (408 pages)
Ausgabe:
4th ed.
ISBN:
9783031410260
Serie:
Undergraduate Texts in Mathematics Series
Anmerkung:
Intro -- About the Author -- Contents -- Preface for Students -- Preface for Instructors -- The author's top ten -- Major improvements and additions for the fourth edition -- Acknowledgments -- Chapter 1 Vector Spaces -- 1A Rⁿ and Cⁿ -- Complex Numbers -- Lists -- Fⁿ -- Digression on Fields -- Exercises 1A -- 1B Definition of Vector Space -- Exercises 1B -- 1C Subspaces -- Sums of Subspaces -- Direct Sums -- Exercises 1C -- Chapter 2 Finite-Dimensional Vector Spaces -- 2A Span and Linear Independence -- Linear Combinations and Span -- Linear Independence -- Exercises 2A -- 2B Bases -- Exercises 2B -- 2C Dimension -- Exercises 2C -- Chapter 3 Linear Maps -- 3A Vector Space of Linear Maps -- Definition and Examples of Linear Maps -- Algebraic Operations on L(V, W) -- Exercises 3A -- 3B Null Spaces and Ranges -- Null Space and Injectivity -- Range and Surjectivity -- Fundamental Theorem of Linear Maps -- Exercises 3B -- 3C Matrices -- Representing a Linear Map by a Matrix -- Addition and Scalar Multiplication of Matrices -- Matrix Multiplication -- Column-Row Factorization and Rank of a Matrix -- Exercises 3C -- 3D Invertibility and Isomorphisms -- Invertible Linear Maps -- Isomorphic Vector Spaces -- Linear Maps Thought of as Matrix Multiplication -- Change of Basis -- Exercises 3D -- 3E Products and Quotients of Vector Spaces -- Products of Vector Spaces -- Quotient Spaces -- Exercises 3E -- 3F Duality -- Dual Space and Dual Map -- Null Space and Range of Dual of Linear Map -- Matrix of Dual of Linear Map -- Exercises 3F -- Chapter 4 Polynomials -- Zeros of Polynomials -- Division Algorithm for Polynomials -- Factorization of Polynomials over C -- Factorization of Polynomials over R -- Exercises 4 -- Chapter 5 Eigenvalues and Eigenvectors -- 5A Invariant Subspaces -- Eigenvalues -- Polynomials Applied to Operators -- Exercises 5A.
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5B The Minimal Polynomial -- Existence of Eigenvalues on Complex Vector Spaces -- Eigenvalues and the Minimal Polynomial -- Eigenvalues on Odd-Dimensional Real Vector Spaces -- Exercises 5B -- 5C Upper-Triangular Matrices -- Exercises 5C -- 5D Diagonalizable Operators -- Diagonal Matrices -- Conditions for Diagonalizability -- Gershgorin Disk Theorem -- Exercises 5D -- 5E Commuting Operators -- Exercises 5E -- Chapter 6 Inner Product Spaces -- 6A Inner Products and Norms -- Inner Products -- Norms -- Exercises 6A -- 6B Orthonormal Bases -- Orthonormal Lists and the Gram-Schmidt Procedure -- Linear Functionals on Inner Product Spaces -- Exercises 6B -- 6C Orthogonal Complements and Minimization Problems -- Orthogonal Complements -- Minimization Problems -- Pseudoinverse -- Exercises 6C -- Chapter 7 Operators on Inner Product Spaces -- 7A Self-Adjoint and Normal Operators -- Adjoints -- Self-Adjoint Operators -- Normal Operators -- Exercises 7A -- 7B Spectral Theorem -- Real Spectral Theorem -- Complex Spectral Theorem -- Exercises 7B -- 7C Positive Operators -- Exercises 7C -- 7D Isometries, Unitary Operators, and Matrix Factorization -- Isometries -- Unitary Operators -- QR Factorization -- Cholesky Factorization -- Exercises 7D -- 7E Singular Value Decomposition -- Singular Values -- SVD for Linear Maps and for Matrices -- Exercises 7E -- 7F Consequences of Singular Value Decomposition -- Norms of Linear Maps -- Approximation by Linear Maps with Lower-Dimensional Range -- Polar Decomposition -- Operators Applied to Ellipsoids and Parallelepipeds -- Volume via Singular Values -- Properties of an Operator as Determined by Its Eigenvalues -- Exercises 7F -- Chapter 8 Operators on Complex Vector Spaces -- 8A Generalized Eigenvectors and Nilpotent Operators -- Null Spaces of Powers of an Operator -- Generalized Eigenvectors -- Nilpotent Operators.
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Exercises 8A -- 8B Generalized Eigenspace Decomposition -- Generalized Eigenspaces -- Multiplicity of an Eigenvalue -- Block Diagonal Matrices -- Exercises 8B -- 8C Consequences of Generalized Eigenspace Decomposition -- Square Roots of Operators -- Jordan Form -- Exercises 8C -- 8D Trace: A Connection Between Matrices and Operators -- Exercises 8D -- Chapter 9 Multilinear Algebra and Determinants -- 9A Bilinear Forms and Quadratic Forms -- Bilinear Forms -- Symmetric Bilinear Forms -- Quadratic Forms -- Exercises 9A -- 9B Alternating Multilinear Forms -- Multilinear Forms -- Alternating Multilinear Forms and Permutations -- Exercises 9B -- 9C Determinants -- Defining the Determinant -- Properties of Determinants -- Exercises 9C -- 9D Tensor Products -- Tensor Product of Two Vector Spaces -- Tensor Product of Inner Product Spaces -- Tensor Product of Multiple Vector Spaces -- Exercises 9D -- Photo Credits -- Symbol Index -- Index -- Colophon: Notes on Typesetting.
Weitere Ausg.:
Print version: Axler, Sheldon Linear Algebra Done Right Cham : Springer International Publishing AG,c2023
Sprache:
Englisch
Schlagwort(e):
Electronic books.
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Electronic books.
URL:
Click here to view book
