Linear Algebra Done Right [[electronic resource] /] / by Sheldon Axler

• Autore: Axler Sheldon
• Pubblicazione: Cham : , : Springer International Publishing : , : Imprint : Springer, , 2015
• Edizione: 3rd ed. 2015.
• Descrizione fisica: 1 online resource : color illustrations
• Disciplina: 510
• Soggetto topico: Matrix theory; Algebra; Linear and Multilinear Algebras, Matrix Theory
• Nota di contenuto: Intro -- Contents -- Preface for the Instructor -- Preface for the Student -- Acknowledgments -- CHAPTER 1 -- Vector Spaces -- 1.A Rn and Cn -- Complex Numbers -- 1.1 Definition -- 1.2 Example -- 1.3 Properties of complex arithmetic -- 1.4 Example -- 1.5 Definition -- 1.6 Notation -- Lists -- 1.7 Example -- 1.8 Definition -- 1.9 Example -- 1.10 Definition -- 1.11 Example -- 1.12 Definition -- 1.13 Commutativity of addition in Fn -- 1.14 Definition -- 1.15 Example -- 1.16 Definition -- 1.17 Definition -- Digression on Fields -- EXERCISES 1.A -- 1.B Definition of Vector Space -- 1.18 Definition -- 1.19 Definition -- 1.20 Definition -- 1.21 Definition -- 1.22 Example -- 1.23 Notation -- 1.24 Example -- 1.25 Unique additive identity -- 1.26 Unique additive inverse -- 1.27 Notation -- 1.28 Notation -- 1.29 The number 0 times a vector -- 1.30 A number times the vector 0 -- 1.31 The number 1 times a vector -- EXERCISES 1.B -- 1.C Subspaces -- 1.32 Definition -- 1.33 Example -- 1.34 Conditions for a subspace -- 1.35 Example -- Sums of Subspaces -- 1.36 Definition -- 1.37 Example -- 1.38 Example -- 1.39 Sum of subspaces is the smallest containing subspace -- Direct Sums -- 1.40 Definition -- 1.41 Example -- 1.42 Example -- 1.43 Example -- 1.44 Condition for a direct sum -- 1.45 Direct sum of two subspaces -- EXERCISES 1.C -- CHAPTER 2 -- Finite-Dimensional Vector Spaces -- 2.1 Notation -- 2.A Span and Linear Independence -- Linear Combinations and Span -- 2.2 Notation -- 2.3 Definition -- 2.4 Example -- 2.5 Definition -- 2.6 Example -- 2.7 Span is the smallest containing subspace -- 2.8 Definition -- 2.9 Example -- 2.10 Definition -- 2.11 Definition -- 2.12 Definition -- 2.13 Definition -- 2.14 Example -- 2.15 Definition -- 2.16 Example -- Linear Independence -- 2.17 Definition -- 2.18 Example -- 2.20 Example -- 2.21 Linear Dependence Lemma -- 2.22. ; 2.23 Length of linearly independent list ≤ length of spanning list -- 2.24 Example -- 2.25 Example -- 2.26 Finite-dimensional subspaces -- EXERCISES 2.A -- 2.B Bases -- 2.27 Definition -- 2.28 Example -- 2.29 Criterion for basis -- 2.30 -- 2.31 Spanning list contains a basis -- 2.32 Basis of finite-dimensional vector space -- 2.33 Linearly independent list extends to a basis -- 2.34 Every subspace of V is part of a direct sum equal to V -- EXERCISES 2.B -- 2.C Dimension -- 2.35 Basis length does not depend on basis -- 2.36 Definition -- 2.37 Example -- 2.38 Dimension of a subspace -- 2.39 Linearly independent list of the right length is a basis -- 2.40 Example -- 2.41 Example -- 2.42 Spanning list of the right length is a basis -- 2.43 Dimension of a sum -- EXERCISES 2.C -- CHAPTER 3 -- Linear Maps -- 3.1 Notation -- 3.A The Vector Space of Linear Maps -- Definition and Examples of Linear Maps -- 3.2 Definition -- 3.3 Notation -- 3.4 Example -- 3.5 Linear maps and basis of domain -- Algebraic Operations on L(V,W) -- 3.7 L(V,W) is a vector space -- 3.8 Definition -- 3.9 Algebraic properties of products of linear maps -- 3.10 Example -- 3.11 Linear maps take 0 to 0 -- EXERCISES 3.A -- 3.B Null Spaces and Ranges -- Null Space and Injectivity -- 3.12 Definition -- 3.13 Example -- 3.14 The null space is a subspace -- 3.15 Definition -- 3.16 Injectivity is equivalent to null space equals {0} -- Range and Surjectivity -- 3.17 Definition -- 3.18 Example -- 3.19 The range is a subspace -- 3.20 Definition -- 3.21 Example -- Fundamental Theorem of Linear Maps -- 3.22 Fundamental Theorem of Linear Maps -- 3.23 A map to a smaller dimensional space is not injective -- 3.24 A map to a larger dimensional space is not surjective -- 3.25 Example -- 3.26 Homogeneous system of linear equations -- 3.27 Example -- 3.28 -- 3.29 Inhomogeneous system of linear equations. ; EXERCISES 3.B -- 3.C Matrices -- Representing a Linear Map by a Matrix -- 3.30 Definition -- 3.31 Example -- 3.32 Definition -- 3.33 Example -- 3.34 Example -- Addition and Scalar Multiplication of Matrices -- 3.35 Definition -- 3.36 The matrix of the sum of linear maps -- 3.37 Definition -- 3.38 The matrix of a scalar times a linear map -- 3.39 Notation -- 3.40 dim Fm,n = mn -- Matrix Multiplication -- 3.41 Definition -- 3.42 Example -- 3.43 The matrix of the product of linear maps -- 3.44 Notation -- 3.45 Example -- 3.46 Example -- 3.48 Example -- 3.49 Column of matrix product equals matrix times column -- 3.50 Example -- 3.51 Example -- 3.52 Linear combination of columns -- EXERCISES 3.C -- 3.D Invertibility and Isomorphic Vector Spaces -- Invertible Linear Maps -- 3.53 Definition -- 3.54 Inverse is unique -- 3.55 Notation -- 3.56 Invertibility is equivalent to injectivity and surjectivity -- 3.57 Example -- Isomorphic Vector Spaces -- 3.58 Definition -- 3.59 Dimension shows whether vector spaces are isomorphic -- 3.60 L(V -- W) and Fm,n are isomorphic -- 3.61 dimL(V -- W) = (dimV)(dimW) -- Linear Maps Thought of as Matrix Multiplication -- 3.62 Definition -- 3.63 Example -- 3.64 M(T ).,k = M(vk). -- 3.65 Linear maps act like matrix multiplication -- 3.66 -- Operators -- 3.67 Definition -- 3.68 Example -- 3.69 Injectivity is equivalent to surjectivity in finite dimensions -- 3.70 Example -- EXERCISES 3.D -- 3.E Products and Quotients of Vector Spaces -- Products of Vector Spaces -- 3.71 Definition -- 3.72 Example -- 3.73 Product of vector spaces is a vector space -- 3.74 Example -- 3.75 Example -- 3.76 Dimension of a product is the sum of dimensions -- Products and Direct Sums -- 3.77 Products and direct sums -- 3.78 A sum is a direct sum if and only if dimensions add up -- Quotients of Vector Spaces -- 3.79 Definition -- 3.80 Example. ; 3.81 Definition -- 3.82 Example -- 3.83 Definition -- 3.84 Example -- 3.85 Two affine subsets parallel to U are equal or disjoint -- 3.86 Definition -- 3.87 Quotient space is a vector space -- 3.88 Definition -- 3.89 Dimension of a quotient space -- 3.90 Definition -- 3.91 Null space and range of T -- EXERCISES 3.E -- 3.F Duality -- The Dual Space and the Dual Map -- 3.92 Definition -- 3.93 Example -- 3.94 Definition -- 3.95 dim V' = dim V -- 3.96 Definition -- 3.97 Example -- 3.98 Dual basis is a basis of the dual space -- 3.99 Definition -- 3.100 Example -- 3.101 Algebraic properties of dual maps -- The Null Space and Range of the Dual of a Linear Map -- 3.102 Definition -- 3.103 Example -- 3.104 Example -- 3.105 The annihilator is a subspace -- 3.106 Dimension of the annihilator -- 3.107 The null space of T' -- 3.108 T' surjective is equivalent to T' injective -- 3.109 The range of T' -- 3.110 T' injective is equivalent to T' surjective -- The Matrix of the Dual of a Linear Map -- 3.111 Definition -- 3.112 Example -- 3.113 The transpose of the product of matrices -- 3.114 The matrix of T' is the transpose of the matrix of T' -- The Rank of a Matrix -- 3.115 Definition -- 3.116 Example -- 3.117 Dimension of range T equals column rank of M(T) -- 3.118 Row rank equals column rank -- 3.119 Definition -- EXERCISES 3.F -- CHAPTER 4 -- Polynomials -- 4.1 Notation -- Complex Conjugate and Absolute Value -- 4.2 Definition -- 4.3 Definition -- 4.4 Example -- 4.5 Properties of complex numbers -- Uniqueness of Coefficients for Polynomials -- 4.6 -- 4.7 If a polynomial is the zero function, then all coefficients are 0 -- The Division Algorithm for Polynomials -- 4.8 Division Algorithm for Polynomials -- Zeros of Polynomials -- 4.9 Definition -- 4.10 Definition -- 4.11 Each zero of a polynomial corresponds to a degree-1 factor. ; 4.12 A polynomial has at most as many zeros as its degree -- Factorization of polynomials over C -- 4.13 Fundamental Theorem of Algebra -- 4.14 Factorization of a polynomial over C -- Factorization of polynomials over R -- 4.15 Polynomials with real coefficients have zeros in pairs -- 4.16 Factorization of a quadratic polynomial -- 4.17 Factorization of a polynomial over R -- EXERCISES 4 -- CHAPTER 5 -- Eigenvalues, Eigenvectors, and Invariant Subspaces -- 5.1 Notation -- 5.A Invariant Subspaces -- 5.2 Definition -- 5.3 Example -- 5.4 Example -- Eigenvalues and Eigenvectors -- 5.5 Definition -- 5.6 Equivalent conditions to be an eigenvalue -- 5.7 Definition -- 5.8 Example -- 5.9 -- 5.10 Linearly independent eigenvectors -- 5.11 -- 5.12 -- 5.13 Number of eigenvalues -- Restriction and Quotient Operators -- 5.14 Definition -- 5.15 Example -- EXERCISES 5.A -- 5.B Eigenvectors and Upper-Triangular Matrices -- Polynomials Applied to Operators -- 5.16 Definition -- 5.17 Definition -- 5.18 Example -- 5.19 Definition -- 5.20 Multiplicative properties -- Existence of Eigenvalues -- 5.21 Operators on complex vector spaces have an eigenvalue -- Upper-Triangular Matrices -- 5.22 Definition -- 5.23 Example -- 5.24 Definition -- 5.25 Definition -- 5.26 Conditions for upper-triangular matrix -- 5.27 Over C, every operator has an upper-triangular matrix -- 5.28 -- 5.29 -- 5.30 Determination of invertibility from upper-triangular matrix -- 5.31 -- 5.32 Determination of eigenvalues from upper-triangular matrix -- 5.33 Example -- EXERCISES 5.B -- 5.C Eigenspaces and Diagonal Matrices -- 5.34 Definition -- 5.35 Example -- 5.36 Definition -- 5.37 Example -- 5.38 Sum of eigenspaces is a direct sum -- 5.39 Definition -- 5.40 Example -- 5.41 Conditions equivalent to diagonalizability -- 5.42 -- 5.43 Example -- 5.44 Enough eigenvalues implies diagonalizability. ; 5.45 Example.
• Sommario/riassunto: 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 finite-dimensional 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. The third edition contains major improvements and revisions throughout the book. More than 300 new exercises have been added since the previous edition. Many new examples have been added to illustrate the key ideas of linear algebra. New topics covered in the book include product spaces, quotient spaces, and dual spaces. Beautiful new formatting creates pages with an unusually pleasant appearance in both print and electronic versions. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. Thus the text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator.
• Titolo autorizzato: Linear Algebra done right
• ISBN: 3-319-11080-2
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910299783303321
• Lo trovi qui: Univ. Federico II
• Serie: Undergraduate Texts in Mathematics, . 0172-6056