| Document Type
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BL
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| Record Number
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809847
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| Doc. No
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b623863
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| Main Entry
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Axler, Sheldon Jay
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| Title & Author
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Linear algebra done right /\ Sheldon Axler.
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| Edition Statement
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Third edition.
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| Series Statement
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Undergraduate texts in mathematics,
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| Page. NO
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xvii, 340 pages :: color illustrations ;; 25 cm.
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| ISBN
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9783319110790
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: 3319110799
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9783319110806
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: 3319110802
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: 9783319110806
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: 9783319307657
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: 3319307657
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| Notes
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Includes indexes.
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| Contents
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Machine generated contents note: 1.Vector Spaces -- 1.A.Rn and Cn -- Complex Numbers -- Lists -- Fn -- Digression on Fields -- Exercises 1.A -- 1.B.Definition of Vector Space -- Exercises 1.B -- 1.C.Subspaces -- Sums of Subspaces -- Direct Sums -- Exercises 1.C -- 2.Finite-Dimensional Vector Spaces -- 2.A.Span and Linear Independence -- Linear Combinations and Span -- Linear Independence -- Exercises 2.A -- 2.B.Bases -- Exercises 2.B -- 2.C.Dimension -- Exercises 2.C -- 3.Linear Maps -- 3.A.The Vector Space of Linear Maps -- Definition and Examples of Linear Maps -- Algebraic Operations on L(V, W) -- Exercises 3.A -- 3.B.Null Spaces and Ranges -- Null Space and Injectivity -- Range and Surjectivity -- Fundamental Theorem of Linear Maps -- Exercises 3.B -- 3.C.Matrices -- Representing a Linear Map by a Matrix -- Addition and Scalar Multiplication of Matrices -- Matrix Multiplication -- Exercises 3.C -- 3.D.Invertibility and Isomorphic Vector Spaces -- Invertible Linear Maps
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Note continued: Isomorphic Vector Spaces -- Linear Maps Thought of as Matrix Multiplication -- Operators -- Exercises 3.D -- 3.E.Products and Quotients of Vector Spaces -- Products of Vector Spaces -- Products and Direct Sums -- Quotients of Vector Spaces -- Exercises 3.E -- 3.F.Duality -- The Dual Space and the Dual Map -- The Null Space and Range of the Dual of a Linear Map -- The Matrix of the Dual of a Linear Map -- The Rank of a Matrix -- Exercises 3.F -- 4.Polynomials -- Complex Conjugate and Absolute Value -- Uniqueness of Coefficients for Polynomials -- The Division Algorithm for Polynomials -- Zeros of Polynomials -- Factorization of Polynomials over C -- Factorization of Polynomials over R -- Exercises 4 -- 5.Eigenvalues, Eigenvectors, and Invariant Subspaces -- 5.A.Invariant Subspaces -- Eigenvalues and Eigenvectors -- Restriction and Quotient Operators -- Exercises 5.A -- 5.B.Eigenvectors and Upper-Triangular Matrices -- Polynomials Applied to Operators
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Note continued: Existence of Eigenvalues -- Upper-Triangular Matrices -- Exercises 5.B -- 5.C.Eigenspaces and Diagonal Matrices -- Exercises 5.C -- 6.Inner Product Spaces -- 6.A.Inner Products and Norms -- Inner Products -- Norms -- Exercises 6.A -- 6.B.Orthonormal Bases -- Linear Functionals on Inner Product Spaces -- Exercises 6.B -- 6.C.Orthogonal Complements and Minimization Problems -- Orthogonal Complements -- Minimization Problems -- Exercises 6.C -- 7.Operators on Inner Product Spaces -- 7.A.Self-Adjoint and Normal Operators -- Adjoints -- Self-Adjoint Operators -- Normal Operators -- Exercises 7.A -- 7.B.The Spectral Theorem -- The Complex Spectral Theorem -- The Real Spectral Theorem -- Exercises 7.B -- 7.C.Positive Operators and Isometries -- Positive Operators -- Isometries -- Exercises 7.C -- 7.D.Polar Decomposition and Singular Value Decomposition -- Polar Decomposition -- Singular Value Decomposition -- Exercises 7.D
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Note continued: 8.Operators on Complex Vector Spaces -- 8.A.Generalized Eigenvectors and Nilpotent Operators -- Null Spaces of Powers of an Operator -- Generalized Eigenvectors -- Nilpotent Operators -- Exercises 8.A -- 8.B.Decomposition of an Operator -- Description of Operators on Complex Vector Spaces -- Multiplicity of an Eigenvalue -- Block Diagonal Matrices -- Square Roots -- Exercises 8.B -- 8.C.Characteristic and Minimal Polynomials -- The Cayley -- Hamilton Theorem -- The Minimal Polynomial -- Exercises 8.C -- 8.D.Jordan Form -- Exercises 8.D -- 9.Operators on Real Vector Spaces -- 9.A.Complexification -- Complexification of a Vector Space -- Complexification of an Operator -- The Minimal Polynomial of the Complexification -- Eigenvalues of the Complexification -- Characteristic Polynomial of the Complexification -- Exercises 9.A -- 9.B.Operators on Real Inner Product Spaces -- Normal Operators on Real Inner Product Spaces
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Note continued: Isometries on Real Inner Product Spaces -- Exercises 9.B -- 10.Trace and Determinant -- 10.A.Trace -- Change of Basis -- Trace: A Connection Between Operators and Matrices -- Exercises 10.A -- 10.B.Determinant -- Determinant of an Operator -- Determinant of a Matrix -- The Sign of the Determinant -- Volume -- Exercises 10.B.
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| Abstract
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"This best-selling textbook for a second course in linear algebra is aimed at undergrad math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra." -- Back cover.
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| Subject
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Algebras, Linear.
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Algebras, Linear-- Study and teaching.
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Algebras, Linear-- Study and teaching
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Algebras, Linear.
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| Subject
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Linjär algebra.
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| Subject
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Mathematics.
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| Subject
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Mathematics.
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| Dewey Classification
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512/.5
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| LC Classification
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QA184.A96 2015
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| NLM classification
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512.5AXL
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