Lectures in abstract algebra. Volume II, Linear algebra / by Nathan Jacobson.
Series The University series in higher mathematicsEditor: Princeton : Van Nostrand, c1953Descripción: xii, 280 p. ; 24 cmOtra clasificación: 08-XXCHAPTER I: FINITE DIMENSIONAL VECTOR SPACES SECTION PAGE 1. Abstract vector spaces [3] 2. Right vector spaces [6] 3. o-modules [7] 4. Linear dependence [9] 5. Invariance of dimensionality [13] 6. Bases and matrices [15] 7. Applications to matrix theory [18] 8. Rank of a set of vectors [22] 9. Factor spaces [25] 10. Algebra of subspaces [25] 11. Independent subspaces, direct sums [28] CHAPTER II: LINEAR TRANSFORMATIONS 1. Definition and examples [31] 2. Compositions of linear transformations [33] 3. The matrix of a linear transformation [36] 4. Compositions of matrices [38] 5. Change of basis. Equivalence and similarity of matrices [41] 6. Rank space and null space of a linear transformation [44] 7. Systems of linear equations [47] 8. Linear transformations in right vector spaces [49] 9. Linear functions [51] 10. Duality between a finite dimensional space and its conjugate space [53] 11. Transpose of a linear transformation [56] 12. Matrices of the transpose [58] 13. Projections [59] CHAPTER III: THE THEORY OF A SINGLE LINEAR TRANSFORMATION 1. The minimum polynomial of a linear transformation [63] 2. Cyclic subspaces [66] 3. Existence of a vector whose order is the minimum polynomial [67] 4. Cyclic linear transformations .............. [69] 5. The Φ[λ]-module determined by a linear transformation [74] 6. Finitely generated o-modules, o, a principal ideal domain [76] 7. Normalization of the generators of F and of R [78] 8. Equivalence of matrices with elements in a principal ideal domain [79] 9. Structure of finitely generated o-modules [85] 10. Invariance theorems [88] 11. Decomposition of a vector space relative to a linear transformation [92] 12. The characteristic and minimum polynomials [98] 13. Direct proof of Theorem 13 [100] 14. Formal properties of the trace and the characteristic polynomial [103] 15. The ring of o-endomorphisms of a cyclic o-module [106] 16. Determination of the ring of o-endomorphisms of a finitely generated o-module, o principal [108] 17. The linear transformations which commute with a given linear transformation [110] 18. The center of the ring B [113] CHAPTER IV: SETS OF LINEAR TRANSFORMATIONS 1. Invariant subspaces [115] 2. Induced linear transformations [117] 3. Composition series [120] 4. Decomposability [122] 5. Complete reducibility [124] 6. Relation to the theory of operator groups and the theory of modules [126] 7. Reducibility, decomposability, complete reducibility for a single linear transformation [128] 8. The primary components of a space relative to a linear transformation [130] 9. Sets of commutative linear transformations [132] CHAPTER V: BILINEAR FORMS 1. Bilinear forms [137] 2. Matrices of a bilinear form [138] 3. Non-degenerate forms [140] 4. Transpose of a linear transformation relative to a pair of bilinear forms [142] 5. Another relation between linear transformations and bilinear forms [145] 6. Scalar products [147] 7. Hermitian scalar products [150] 8. Matrices of hermitian scalar products [152] 9. Symmetric and hermitian scalar products over special division rings [154] 10. Alternate scalar products [159] 11. Witt’s theorem [162] 12. Non-alternate skew-symmetric forms [170] CHAPTER VI: EUCLIDEAN AND UNITARY SPACES 1. Cartesian bases [173] 2. Linear transformations and scalar products [176] 3. Orthogonal complete reducibility [177] 4. Symmetric, skew and orthogonal linear transformations [178] 5. Canonical matrices for symmetric and skew linear transformations [179] 6. Commutative symmetric and skew linear transformations [182] 7. Normal and orthogonal linear transformations [184] 8. Semi-definite transformations [186] 9. Polar factorization of an arbitrary linear transformation [188] 10. Unitary geometry [190] 11. Analytic functions of linear transformations [194] CHAPTER VII: PRODUCTS OF VECTOR SPACES 1. Product groups of vector spaces [199] 2. Direct products of linear transformations [202] 3. Two-sided vector spaces [204] 4. The Kronecker product [208] 5. Kronecker products of linear transformations and of matrices [211] 6. Tensor spaces [213] 7. Symmetry classes of tensors [217] 8. Extension of the field of a vector space [221] 9. A theorem on similarity of sets of matrices [222] 10. Alternative definition of an algebra. Kronecker product of algebras [225] CHAPTER VIII: THE RING OF LINEAR TRANSFORMATIONS 1. Simplicity of [227] 2. Operator methods [229] 3. The left ideals of [230] 4. Right ideals [232] 5. Isomorphisms of rings of linear transformations [233] CHAPTER IX: INFINITE DIMENSIONAL VECTOR SPACES 1. Existence of a basis [239] 2. Invariance of dimensionality [240] 3. Subspaces [242] 4. Linear transformations and matrices [243] 5. Dimensionality of the conjugate space [244] 6. Finite topology for linear transformations [248] 7. Total subspaces of R* [251] 8. Dual spaces. Kronecker products [253] 9. Two-sided ideals in the ring of linear transformations [256] 10. Dense rings of linear transformations [259] 11. Isomorphism theorems [264] 12. Anti-automorphisms and scalar products [267] 13. Schur’s lemma. A general density theorem [271] 14. Irreducible algebras of linear transformations [274]
| Item type | Home library | Shelving location | Call number | Materials specified | Copy number | Status | Date due | Barcode | Course reserves |
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 08 J17 (Browse shelf) | Vol. II | Available | A-72 | |||
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 08 J17 (Browse shelf) | Vol. II | Ej. 2 | Available | A-2337 |
CHAPTER I: FINITE DIMENSIONAL VECTOR SPACES SECTION PAGE --
1. Abstract vector spaces [3] --
2. Right vector spaces [6] --
3. o-modules [7] --
4. Linear dependence [9] --
5. Invariance of dimensionality [13] --
6. Bases and matrices [15] --
7. Applications to matrix theory [18] --
8. Rank of a set of vectors [22] --
9. Factor spaces [25] --
10. Algebra of subspaces [25] --
11. Independent subspaces, direct sums [28] --
CHAPTER II: LINEAR TRANSFORMATIONS --
1. Definition and examples [31] --
2. Compositions of linear transformations [33] --
3. The matrix of a linear transformation [36] --
4. Compositions of matrices [38] --
5. Change of basis. Equivalence and similarity of matrices [41] --
6. Rank space and null space of a linear transformation [44] --
7. Systems of linear equations [47] --
8. Linear transformations in right vector spaces [49] --
9. Linear functions [51] --
10. Duality between a finite dimensional space and its conjugate space [53] --
11. Transpose of a linear transformation [56] --
12. Matrices of the transpose [58] --
13. Projections [59] --
CHAPTER III: THE THEORY OF A SINGLE LINEAR TRANSFORMATION --
1. The minimum polynomial of a linear transformation [63] --
2. Cyclic subspaces [66] --
3. Existence of a vector whose order is the minimum polynomial [67] --
4. Cyclic linear transformations .............. [69] --
5. The Φ[λ]-module determined by a linear transformation [74] --
6. Finitely generated o-modules, o, a principal ideal domain [76] --
7. Normalization of the generators of F and of R [78] --
8. Equivalence of matrices with elements in a principal ideal domain [79] --
9. Structure of finitely generated o-modules [85] --
10. Invariance theorems [88] --
11. Decomposition of a vector space relative to a linear transformation [92] --
12. The characteristic and minimum polynomials [98] --
13. Direct proof of Theorem 13 [100] --
14. Formal properties of the trace and the characteristic polynomial [103] --
15. The ring of o-endomorphisms of a cyclic o-module [106] --
16. Determination of the ring of o-endomorphisms of a finitely generated o-module, o principal [108] --
17. The linear transformations which commute with a given linear transformation [110] --
18. The center of the ring B [113] --
CHAPTER IV: SETS OF LINEAR TRANSFORMATIONS --
1. Invariant subspaces [115] --
2. Induced linear transformations [117] --
3. Composition series [120] --
4. Decomposability [122] --
5. Complete reducibility [124] --
6. Relation to the theory of operator groups and the theory of modules [126] --
7. Reducibility, decomposability, complete reducibility for a single linear transformation [128] --
8. The primary components of a space relative to a linear transformation [130] --
9. Sets of commutative linear transformations [132] --
CHAPTER V: BILINEAR FORMS --
1. Bilinear forms [137] --
2. Matrices of a bilinear form [138] --
3. Non-degenerate forms [140] --
4. Transpose of a linear transformation relative to a pair of bilinear forms [142] --
5. Another relation between linear transformations and bilinear forms [145] --
6. Scalar products [147] --
7. Hermitian scalar products [150] --
8. Matrices of hermitian scalar products [152] --
9. Symmetric and hermitian scalar products over special division rings [154] --
10. Alternate scalar products [159] --
11. Witt’s theorem [162] --
12. Non-alternate skew-symmetric forms [170] --
CHAPTER VI: EUCLIDEAN AND UNITARY SPACES --
1. Cartesian bases [173] --
2. Linear transformations and scalar products [176] --
3. Orthogonal complete reducibility [177] --
4. Symmetric, skew and orthogonal linear transformations [178] --
5. Canonical matrices for symmetric and skew linear transformations [179] --
6. Commutative symmetric and skew linear transformations [182] --
7. Normal and orthogonal linear transformations [184] --
8. Semi-definite transformations [186] --
9. Polar factorization of an arbitrary linear transformation [188] --
10. Unitary geometry [190] --
11. Analytic functions of linear transformations [194] --
CHAPTER VII: PRODUCTS OF VECTOR SPACES --
1. Product groups of vector spaces [199] --
2. Direct products of linear transformations [202] --
3. Two-sided vector spaces [204] --
4. The Kronecker product [208] --
5. Kronecker products of linear transformations and of matrices [211] --
6. Tensor spaces [213] --
7. Symmetry classes of tensors [217] --
8. Extension of the field of a vector space [221] --
9. A theorem on similarity of sets of matrices [222] --
10. Alternative definition of an algebra. Kronecker product of algebras [225] --
CHAPTER VIII: THE RING OF LINEAR TRANSFORMATIONS --
1. Simplicity of [227] --
2. Operator methods [229] --
3. The left ideals of [230] --
4. Right ideals [232] --
5. Isomorphisms of rings of linear transformations [233] --
CHAPTER IX: INFINITE DIMENSIONAL VECTOR SPACES --
1. Existence of a basis [239] --
2. Invariance of dimensionality [240] --
3. Subspaces [242] --
4. Linear transformations and matrices [243] --
5. Dimensionality of the conjugate space [244] --
6. Finite topology for linear transformations [248] --
7. Total subspaces of R* [251] --
8. Dual spaces. Kronecker products [253] --
9. Two-sided ideals in the ring of linear transformations [256] --
10. Dense rings of linear transformations [259] --
11. Isomorphism theorems [264] --
12. Anti-automorphisms and scalar products [267] --
13. Schur’s lemma. A general density theorem [271] --
14. Irreducible algebras of linear transformations [274] --
MR, 14,837e
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