| Autore |
Axler Sheldon
|
| Edizione | [3rd ed. 2015.] |
| Pubbl/distr/stampa |
Cham : , : Springer International Publishing : , : Imprint : Springer, , 2015
|
| Descrizione fisica |
1 online resource : color illustrations
|
| Disciplina |
510
|
| Collana |
Undergraduate Texts in Mathematics
|
| Soggetto topico |
Matrix theory
Algebra
Linear and Multilinear Algebras, Matrix Theory
|
| ISBN |
3-319-11080-2
|
| Formato |
Materiale a stampa  |
| Livello bibliografico |
Monografia |
| Lingua di pubblicazione |
eng
|
| 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.
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| Record Nr. | UNINA-9910299783303321 |
Info
Linear Algebra Done Right [[electronic resource] /] / by Sheldon Axler
