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E-raamat: Algebra 2: Linear Algebra, Galois Theory, Representation theory, Group extensions and Schur Multiplier

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Algebra 2: Linear Algebra, Galois Theory, Representation theory, Group extensions and Schur MultiplierSuurem pilt
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This is the second in a series of three volumes dealing with important topics in algebra. Volume 2 is an introduction to linear algebra (including linear algebra over rings), Galois theory, representation theory, and the theory of group extensions. The section on linear algebra (chapters 1–5) does not require any background material from Algebra 1, except an understanding of set theory. Linear algebra is the most applicable branch of mathematics, and it is essential for students of science and engineering As such, the text can be used for one-semester courses for these students. The remaining part of the volume discusses Jordan and rational forms, general linear algebra (linear algebra over rings), Galois theory, representation theory (linear algebra over group algebras), and the theory of extension of groups follow linear algebra, and is suitable as a text for the second and third year students specializing in mathematics.

Arvustused

The text is enhanced by a large number of examples and exercises, and the presentation of the material is equally lucid, detailed, rigorous and versatile. Together with Volume 1, this book forms a very solid and useful source for a first-, second-, and third-year course in algebra at most universities worldwide, and that for both instructors and students likewise. (Werner Kleinert, zbMATH 1369.00003, 2017)

1 Vector Spaces
1(30)
1.1 Concept of a Field
1(6)
1.2 Concept of a Vector Space (Linear Space)
7(4)
1.3 Subspaces
11(5)
1.4 Basis and Dimension
16(7)
1.5 Direct Sum of Vector Spaces, Quotient of a Vector Space
23(8)
2 Matrices and Linear Equations
31(42)
2.1 Matrices and Their Algebra
31(4)
2.2 Types of Matrices
35(5)
2.3 System of Linear Equations
40(3)
2.4 Gauss Elimination, Elementary Operations, Rank, and Nullity
43(15)
2.5 LU Factorization
58(2)
2.6 Equivalence of Matrices, Normal Form
60(5)
2.7 Congruent Reduction of Symmetric Matrices
65(8)
3 Linear Transformations
73(24)
3.1 Definition and Examples
73(2)
3.2 Isomorphism Theorems
75(4)
3.3 Space of Linear Transformations, Dual Spaces
79(4)
3.4 Rank and Nullity
83(2)
3.5 Matrix Representations of Linear Transformations
85(3)
3.6 Effect of Change of Bases on Matrix Representation
88(9)
4 Inner Product Spaces
97(34)
4.1 Definition, Examples, and Basic Properties
97(10)
4.2 Gram--Schmidt Process
107(5)
4.3 Orthogonal Projection, Shortest Distance
112(8)
4.4 Isometries and Rigid Motions
120(11)
5 Determinants and Forms
131(64)
5.1 Determinant of a Matrix
131(4)
5.2 Permutations
135(4)
5.3 Alternating Forms, Determinant of an Endomorphism
139(11)
5.4 Invariant Subspaces, Eigenvalues
150(9)
5.5 Spectral Theorem, and Orthogonal Reduction
159(17)
5.6 Bilinear and Quadratic Forms
176(19)
6 Canonical Forms, Jordan and Rational Forms
195(34)
6.1 Concept of a Module over a Ring
195(8)
6.2 Modules over P.I.D.
203(11)
6.3 Rational and Jordan Forms
214(15)
7 General Linear Algebra
229(36)
7.1 Noetherian Rings and Modules
229(5)
7.2 Free, Projective, and Injective Modules
234(16)
7.3 Tensor Product and Exterior Power
250(8)
7.4 Lower K-theory
258(7)
8 Field Theory, Galois Theory
265(66)
8.1 Field Extensions
265(10)
8.2 Galois Extensions
275(9)
8.3 Splitting Field, Normal Extensions
284(10)
8.4 Separable Extensions
294(11)
8.5 Fundamental Theorem of Galois Theory
305(6)
8.6 Cyclotomic Extensions
311(7)
8.7 Geometric Constructions
318(6)
8.8 Galois Theory of Equation
324(7)
9 Representation Theory of Finite Groups
331(36)
9.1 Semi-simple Rings and Modules
331(15)
9.2 Representations and Group Algebras
346(5)
9.3 Characters, Orthogonality Relations
351(10)
9.4 Induced Representations
361(6)
10 Group Extensions and Schur Multiplier
367(60)
10.1 Schreier Group Extensions
368(23)
10.2 Obstructions and Extensions
391(7)
10.3 Central Extensions, Schur Multiplier
398(20)
10.4 Lower K-Theory Revisited
418(9)
Bibliography 427(2)
Index 429
RAMJI LAL is an adjunct professor at the Harish-Chandra Research Institute (HRI), Allahabad, Uttar Pradesh. He started his research career at the Tata Institute of Fundamental Research (TIFR), Mumbai, and served the University of Allahabad in different capacities for over 43 years: as a professor, head of the department and the coordinator of the DSA program. He was associated with HRI, where he initiated a postgraduate (PG) program in mathematics and coordinated the Nurture Program of National Board for Higher Mathematics (NBHM) from 1996 to 2000. After his retirement from the University of Allahabad, he was an advisor cum adjunct professor at the Indian Institute of Information Technology (IIIT), Allahabad for over three years. His areas of interest include group theory, algebraic K-theory and representation theory.
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