Linear algebra / Sterling K. Berberian.
Editor: Oxford : Oxford University Press, 1992Descripción: xiv, 356 p. : il. ; 24 cmISBN: 0198534361; 0198534353 (pbk.)Tema(s): Algebras, LinearOtra clasificación: 15-01PART I 1 Vector spaces [3] 1.1 Motivation (vectors in 3-space) [3] 1.2 Rn and Cn [8] 1.3 Vector spaces: the axioms, some examples [10] 1.4 Vector spaces: first consequences of the axioms [14] 1.5 Linear combinations of vectors [17] 1.6 Linear subspaces [19] 2 Linear mappings 2.1 Linear mappings [25] 2.2 Linear mappings and linear subspaces: [29] kernel and range [32] 2.3 Spaces of linear mappings: L(V, W) and L(V) [37] 2.4 Isomorphic vector spaces [40] 2.5 Equivalence relations and quotient sets [46] 2.6 Quotient vector spaces [49] 2.7 The first isomorphism theorem [52] 3 Structure of vector spaces 3.1 Linear subspace generated by a subset [53] 3.2 Linear dependence [55] 3.3 Linear independence [58] 3.4 Finitely generated vector spaces [63] 3.5 Basis, dimension [65] 3.6 Rank + nullity = dimension [74] 3.7 Applications of R + N = D [77] 3.8 Dimension of L(V, W) [81] 3.9 Duality in vector spaces [85] 4 Matrices [92] 4.1 Matrices [92] 4.2 Matrices of linear mappings [96] 4.3 Matrix multiplication [102] 4.4 Algebra of matrices [105] 4.5 A model for linear mappings [108] 4.6 Transpose of a matrix [110] 4.7 Calculating the rank [114] 4.8 When is a linear system solvable? [120] 4.9 An example [122] 4.10 Change of basis, similar matrices [125] 5 Inner product spaces [131] 5.1 Inner product spaces, Euclidean spaces [131] 5.2 Duality in inner product spaces [136] 5.3 The adjoint of a linear mapping [143] 5.4 Orthogonal mappings and matrices [147] 6 Determinants (2x2 and 3x3) [154] 6.1 Determinant of a 2 x 2 matrix [155] 6.2 Cross product of vectors in R3 [158] 6.3 Determinant of a 3 x 3 matrix [162] 6.4 Characteristic polynomial of a matrix (2 x 2 or 3 x 3) [165] 6.5 Diagonalizing 2x2 symmetric real matrices [173] 6.6 Diagonalizing 3x3 symmetric real matrices [177] 6.7 A geometric application (conic sections) [182] PART II 7 Determinants (n x n) [193] 7.1 Alternate multilinear forms [193] 7.2 Determinant of a linear mapping [199] 7.3 Determinant of a square matrix [202] 7.4 Cofactors [205] 8 Similarity (Act I) [212] 8.1 Similarity [212] 8.2 Eigenvalues and eigenvectors [215] 8.3 Characteristic polynomial [218] 9 Euclidean spaces (Spectral Theory) [226] 9.1 Invariant and reducing subspaces [226] 9.2 Bounds of a linear mapping [231] 9.3 Bounds of a self-adjoint mapping, Spectral Theorem [235] 9.4 Normal linear mappings in Euclidean spaces [238] 10 Equivalence of matrices over a PIR [244] 10.1 Unimodular matrices [244] 10.2 Preview of the theory of equivalence [246] 10.3 Equivalence: existence of a diagonal form [248] 10.4 Equivalence: uniqueness of the diagonal form [255] 11 Similarity (Act II) [261] 11.1 Invariant factors, Fundamental theorem of similarity [261] 11.2 Companion matrix, Rational canonical form [269] 11.3 Hamilton-Cayley theorem, minimal polynomial [273] 11.4 Elementary divisors, Jordan canonical form [279] 11.5 Appendix: proof that Mn(F)[t] = Mn(F[t]) [284] 12 Unitary spaces [286] 12.1 Complex inner product spaces, unitary spaces [286] 12.2 Orthogonality [290] 12.3 Orthonormal bases, isomorphism [292] 12.4 Adjoint of a linear mapping [294] 12.5 Invariant and reducing subspaces [298] 12.6 Special linear mappings and matrices [299] 12.7 Normal linear mappings, Spectral Theorem [301] 12.8 The Spectral Theorem: another way [301] 13 Tensor products [317] 13.1 Tensor product V x W of vector spaces [317] 13.2 Tensor product S xT of linear mappings [318] 13.3 Matrices of tensor products [322] Appendix A Foundations [326] A. I A dab of logic [326] A.2 Set notations [332] A.3 Functions [335] A.4 The axioms for a field [337] Appendix B Integral domains, factorization theory [338] B.I The field of fractions of an integral domain [338] B.2 Divisibility in an integral domain [340] B.3 Principal ideal rings [343] B.4 Euclidean integral domains [346] B.5 Factorization in overfields [347] Appendix C Weierstrass-Bolzano theorem [350] Index of notations [353] Index [355]
| Item type | Home library | Call number | Materials specified | Copy number | Status | Date due | Barcode | Course reserves |
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Instituto de Matemática, CONICET-UNS | 15 B484 (Browse shelf) | Available | A-6847 | ||||
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Instituto de Matemática, CONICET-UNS | 15 B484 (Browse shelf) | Ej. 2 | Available | A-6848 |
Incluye referencias bibliográficas e índices.
PART I --
1 Vector spaces [3] --
1.1 Motivation (vectors in 3-space) [3] --
1.2 Rn and Cn [8] --
1.3 Vector spaces: the axioms, some examples [10] --
1.4 Vector spaces: first consequences of the axioms [14] --
1.5 Linear combinations of vectors [17] --
1.6 Linear subspaces [19] --
2 Linear mappings --
2.1 Linear mappings [25] --
2.2 Linear mappings and linear subspaces: [29] --
kernel and range [32] --
2.3 Spaces of linear mappings: L(V, W) and L(V) [37] --
2.4 Isomorphic vector spaces [40] --
2.5 Equivalence relations and quotient sets [46] --
2.6 Quotient vector spaces [49] --
2.7 The first isomorphism theorem [52] --
3 Structure of vector spaces --
3.1 Linear subspace generated by a subset [53] --
3.2 Linear dependence [55] --
3.3 Linear independence [58] --
3.4 Finitely generated vector spaces [63] --
3.5 Basis, dimension [65] --
3.6 Rank + nullity = dimension [74] --
3.7 Applications of R + N = D [77] --
3.8 Dimension of L(V, W) [81] --
3.9 Duality in vector spaces [85] --
4 Matrices [92] --
4.1 Matrices [92] --
4.2 Matrices of linear mappings [96] --
4.3 Matrix multiplication [102] --
4.4 Algebra of matrices [105] --
4.5 A model for linear mappings [108] --
4.6 Transpose of a matrix [110] --
4.7 Calculating the rank [114] --
4.8 When is a linear system solvable? [120] --
4.9 An example [122] --
4.10 Change of basis, similar matrices [125] --
5 Inner product spaces [131] --
5.1 Inner product spaces, Euclidean spaces [131] --
5.2 Duality in inner product spaces [136] --
5.3 The adjoint of a linear mapping [143] --
5.4 Orthogonal mappings and matrices [147] --
6 Determinants (2x2 and 3x3) [154] --
6.1 Determinant of a 2 x 2 matrix [155] --
6.2 Cross product of vectors in R3 [158] --
6.3 Determinant of a 3 x 3 matrix [162] --
6.4 Characteristic polynomial of a matrix (2 x 2 or 3 x 3) [165] --
6.5 Diagonalizing 2x2 symmetric real matrices [173] --
6.6 Diagonalizing 3x3 symmetric real matrices [177] --
6.7 A geometric application (conic sections) [182] --
PART II 7 Determinants (n x n) [193] --
7.1 Alternate multilinear forms [193] --
7.2 Determinant of a linear mapping [199] --
7.3 Determinant of a square matrix [202] --
7.4 Cofactors [205] --
8 Similarity (Act I) [212] --
8.1 Similarity [212] --
8.2 Eigenvalues and eigenvectors [215] --
8.3 Characteristic polynomial [218] --
9 Euclidean spaces (Spectral Theory) [226] --
9.1 Invariant and reducing subspaces [226] --
9.2 Bounds of a linear mapping [231] --
9.3 Bounds of a self-adjoint mapping, Spectral Theorem [235] --
9.4 Normal linear mappings in Euclidean spaces [238] --
10 Equivalence of matrices over a PIR [244] --
10.1 Unimodular matrices [244] --
10.2 Preview of the theory of equivalence [246] --
10.3 Equivalence: existence of a diagonal form [248] --
10.4 Equivalence: uniqueness of the diagonal form [255] --
11 Similarity (Act II) [261] --
11.1 Invariant factors, Fundamental theorem of similarity [261] --
11.2 Companion matrix, Rational canonical form [269] --
11.3 Hamilton-Cayley theorem, minimal polynomial [273] --
11.4 Elementary divisors, Jordan canonical form [279] --
11.5 Appendix: proof that Mn(F)[t] = Mn(F[t]) [284] --
12 Unitary spaces [286] --
12.1 Complex inner product spaces, unitary spaces [286] --
12.2 Orthogonality [290] --
12.3 Orthonormal bases, isomorphism [292] --
12.4 Adjoint of a linear mapping [294] --
12.5 Invariant and reducing subspaces [298] --
12.6 Special linear mappings and matrices [299] --
12.7 Normal linear mappings, Spectral Theorem [301] --
12.8 The Spectral Theorem: another way [301] --
13 Tensor products [317] --
13.1 Tensor product V x W of vector spaces [317] --
13.2 Tensor product S xT of linear mappings [318] --
13.3 Matrices of tensor products [322] --
Appendix A Foundations [326] --
A. I A dab of logic [326] --
A.2 Set notations [332] --
A.3 Functions [335] --
A.4 The axioms for a field [337] --
Appendix B Integral domains, factorization theory [338] --
B.I The field of fractions of an integral domain [338] --
B.2 Divisibility in an integral domain [340] --
B.3 Principal ideal rings [343] --
B.4 Euclidean integral domains [346] --
B.5 Factorization in overfields [347] --
Appendix C Weierstrass-Bolzano theorem [350] --
Index of notations [353] --
Index [355] --
MR, REVIEW #
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