Algebra 2: Linear Algebra, Galois Theory, Representation theory, Group extensions and Schur Multiplier (Infosys Science Foundation Series)

Main subject categories: • Linear algebra • Linear transformations • Inner product spaces • Determinants • Field theory • Galois theory • Representation theory • Group extensions • Schur multiplierThis is the second in a series of 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. Preface 7 Contents 9 About the Author 11 Notations from Algebra 1 12 Notations from Algebra 2 15 1 Vector Spaces 17 1.1 Concept of a Field 17 1.2 Concept of a Vector Space (Linear Space) 23 1.3 Subspaces 27 1.4 Basis and Dimension 32 1.5 Direct Sum of Vector Spaces, Quotient of a Vector Space 39 2 Matrices and Linear Equations 47 2.1 Matrices and Their Algebra 47 2.2 Types of Matrices 51 2.3 System of Linear Equations 56 2.4 Gauss Elimination, Elementary Operations, Rank, and Nullity 59 2.5 LU Factorization 74 2.6 Equivalence of Matrices, Normal Form 76 2.7 Congruent Reduction of Symmetric Matrices 81 3 Linear Transformations 88 3.1 Definition and Examples 88 3.2 Isomorphism Theorems 90 3.3 Space of Linear Transformations, Dual Spaces 94 3.4 Rank and Nullity 98 3.5 Matrix Representations of Linear Transformations 100 3.6 Effect of Change of Bases on Matrix Representation 103 4 Inner Product Spaces 111 4.1 Definition, Examples, and Basic Properties 111 4.2 Gram--Schmidt Process 121 4.3 Orthogonal Projection, Shortest Distance 126 4.4 Isometries and Rigid Motions 134 5 Determinants and Forms 144 5.1 Determinant of a Matrix 144 5.2 Permutations 148 5.3 Alternating Forms, Determinant of an Endomorphism 152 5.4 Invariant Subspaces, Eigenvalues 163 5.5 Spectral Theorem, and Orthogonal Reduction 172 5.6 Bilinear and Quadratic Forms 189 6 Canonical Forms, Jordan and Rational Forms 207 6.1 Concept of a Module over a Ring 207 6.2 Modules over P.I.D 215 6.3 Rational and Jordan Forms 226 7 General Linear Algebra 241 7.1 Noetherian Rings and Modules 241 7.2 Free, Projective, and Injective Modules 246 7.3 Tensor Product and Exterior Power 262 7.4 Lower K-theory 270 8 Field Theory, Galois Theory 276 8.1 Field Extensions 276 8.2 Galois Extensions 286 8.3 Splitting Field, Normal Extensions 295 8.4 Separable Extensions 305 8.5 Fundamental Theorem of Galois Theory 316 8.6 Cyclotomic Extensions 322 8.7 Geometric Constructions 329 8.8 Galois Theory of Equation 335 9 Representation Theory of Finite Groups 341 9.1 Semi-simple Rings and Modules 341 9.2 Representations and Group Algebras 356 9.3 Characters, Orthogonality Relations 361 9.4 Induced Representations 371 10 Group Extensions and Schur Multiplier 377 10.1 Schreier Group Extensions 378 10.2 Obstructions and Extensions 401 10.3 Central Extensions, Schur Multiplier 408 10.4 Lower K-Theory Revisited 428 Bibliography 436 Index 437 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. Front Matter....Pages i-xviii Vector Spaces....Pages 1-30 Matrices and Linear Equations....Pages 31-71 Linear Transformations....Pages 73-95 Inner Product Spaces....Pages 97-129 Determinants and Forms....Pages 131-193 Canonical Forms, Jordan and Rational Forms....Pages 195-228 General Linear Algebra....Pages 229-263 Field Theory, Galois Theory....Pages 265-329 Representation Theory of Finite Groups....Pages 331-366 Group Extensions and Schur Multiplier....Pages 367-425 Back Matter....Pages 427-432