Muutke küpsiste eelistusi

E-raamat: Groups, Matrices, and Vector Spaces: A Group Theoretic Approach to Linear Algebra

  • Formaat: EPUB+DRM
  • Ilmumisaeg: 02-Sep-2017
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9780387794280
Teised raamatud teemal:
  • Formaat - EPUB+DRM
  • Hind: 55,56 €*
  • * hind on lõplik, st. muud allahindlused enam ei rakendu
  • Lisa ostukorvi
  • Lisa soovinimekirja
  • See e-raamat on mõeldud ainult isiklikuks kasutamiseks. E-raamatuid ei saa tagastada.
Groups, Matrices, and Vector Spaces: A Group Theoretic Approach to Linear AlgebraSuurem pilt
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 02-Sep-2017
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9780387794280
Teised raamatud teemal:

DRM piirangud

  • Kopeerimine (copy/paste):

    ei ole lubatud

  • Printimine:

    ei ole lubatud

  • Kasutamine:

    Digitaalõiguste kaitse (DRM)
    Kirjastus on väljastanud selle e-raamatu krüpteeritud kujul, mis tähendab, et selle lugemiseks peate installeerima spetsiaalse tarkvara. Samuti peate looma endale  Adobe ID Rohkem infot siin. E-raamatut saab lugeda 1 kasutaja ning alla laadida kuni 6'de seadmesse (kõik autoriseeritud sama Adobe ID-ga).

    Vajalik tarkvara
    Mobiilsetes seadmetes (telefon või tahvelarvuti) lugemiseks peate installeerima selle tasuta rakenduse: PocketBook Reader (iOS / Android)

    PC või Mac seadmes lugemiseks peate installima Adobe Digital Editionsi (Seeon tasuta rakendus spetsiaalselt e-raamatute lugemiseks. Seda ei tohi segamini ajada Adober Reader'iga, mis tõenäoliselt on juba teie arvutisse installeeritud )

    Seda e-raamatut ei saa lugeda Amazon Kindle's. 

This unique text provides a geometric approach to group theory and linear algebra, bringing to light the interesting ways in which these subjects interact. Requiring few prerequisites beyond understanding the notion of a proof, the text aims to give students a strong foundation in both geometry and algebra. Starting with preliminaries (relations, elementary combinatorics, and induction), the book then proceeds to the core topics: the elements of the theory of groups and fields (Lagrange's Theorem, cosets, the complex numbers and the prime fields), matrix theory and matrix groups, determinants, vector spaces, linear mappings, eigentheory and diagonalization, Jordan decomposition and normal form, normal matrices, and quadratic forms. The final two chapters consist of a more intensive look at group theory, emphasizing orbit stabilizer methods, and an introduction to linear algebraic groups, which enriches the notion of a matrix group.

Applications involving symmetry groups, determinants, linear coding theory and cryptography are interwoven throughout. Each section ends with ample practice problems assisting the reader to better understand the material.  Some of the applications are illustrated in the chapter appendices. The author's unique melding of topics evolved from a two semester course that he taught at the University of British Columbia consisting of an undergraduate honors course on abstract linear algebra and a similar course on the theory of groups. The combined content from both makes this rare text ideal for a year-long course, covering more material than most linear algebra texts. It is also optimal for independent study and as a supplementary text for various professional applications. Advanced undergraduate or graduate students in mathematics, physics, computer science and engineering will find this book both useful and enjoyable.

Arvustused

This is an introductory text on linear algebra and group theory from a geometric viewpoint. The topics, largely standard, are presented in brief, well-organized one- and two-page subsections written in clear, if rather pedestrian, language, with detailed examples. (R. J. Bumcrot, Mathematical Reviews, February, 2018)

It is particularly applicable for anyone who is familiar with vector spaces and wants to learn about groups and also for anyone who is familiar with groups and wants to learn about vector spaces. This book is well readable and therefore suitable for self-studying. Each chapter begins with a concise and informative summary of its content, guiding the reader to choose the chapters with most interest to him/her. (Jorma K. Merikoski, zbMATH 1380.15001, 2018)

1 Preliminaries
1(10)
1.1 Sets and Mappings
1(5)
1.1.1 Binary operations
2(2)
1.1.2 Equivalence relations and equivalence classes
4(2)
1.2 Some Elementary Combinatorics
6(5)
1.2.1 Mathematical induction
7(1)
1.2.2 The Binomial Theorem
8(3)
2 Groups and Fields: The Two Fundamental Notions of Algebra
11(46)
2.1 Groups and homomorphisms
11(12)
2.1.1 The Definition of a Group
12(1)
2.1.2 Some basic properties of groups
13(1)
2.1.3 The symmetric groups S(n)
14(1)
2.1.4 Cyclic groups
15(1)
2.1.5 Dihedral groups: generators and relations
16(2)
2.1.6 Subgroups
18(1)
2.1.7 Homomorphisms and Cayley's Theorem
19(4)
2.2 The Cosets of a Subgroup and Lagrange's Theorem
23(6)
2.2.1 The definition of a coset
23(2)
2.2.2 Lagrange's Theorem
25(4)
2.3 Normal Subgroups and Quotient Groups
29(7)
2.3.1 Normal subgroups
29(1)
2.3.2 Constructing the quotient group G/H
30(2)
2.3.3 Euler's Theorem via quotient groups
32(2)
2.3.4 The First Isomorphism Theorem
34(2)
2.4 Fields
36(4)
2.4.1 The definition of a field
36(2)
2.4.2 Arbitrary sums and products
38(2)
2.5 The Basic Number Fields Q, R, and C
40(7)
2.5.1 The rational numbers Q
40(1)
2.5.2 The real numbers R
40(1)
2.5.3 The complex numbers C
41(2)
2.5.4 The geometry of C
43(2)
2.5.5 The Fundamental Theorem of Algebra
45(2)
2.6 Galois fields
47(10)
2.6.1 The prime fields Fp
47(1)
2.6.2 A four-element field
48(1)
2.6.3 The characteristic of a field
49(2)
2.6.4 Appendix: polynomials over a field
51(6)
3 Matrices
57(28)
3.1 Introduction to matrices and matrix algebra
57(11)
3.1.1 What is a matrix?
58(1)
3.1.2 Matrix addition
59(1)
3.1.3 Examples: matrices over F2
60(1)
3.1.4 Matrix multiplication
61(2)
3.1.5 The Algebra of Matrix Multiplication
63(1)
3.1.6 The transpose of a matrix
64(1)
3.1.7 Matrices and linear mappings
65(3)
3.2 Reduced Row Echelon Form
68(9)
3.2.1 Reduced row echelon form and row operations
68(2)
3.2.2 Elementary matrices and row operations
70(2)
3.2.3 The row space and uniqueness of reduced row echelon form
72(5)
3.3 Linear Systems
77(8)
3.3.1 The coefficient matrix of a linear system
77(1)
3.3.2 Writing the solutions: the homogeneous case
78(1)
3.3.3 The inhomogeneous case
79(3)
3.3.4 A useful identity
82(3)
4 Matrix Inverses, Matrix Groups and the LPDU Decomposition
85(28)
4.1 The Inverse of a Square Matrix
85(8)
4.1.1 The definition of the inverse
85(1)
4.1.2 Results on Inverses
86(2)
4.1.3 Computing inverses
88(5)
4.2 Matrix Groups
93(7)
4.2.1 The definition of a matrix group
93(1)
4.2.2 Examples of matrix groups
94(1)
4.2.3 The group of permutation matrices
95(5)
4.3 The LPDU Factorization
100(13)
4.3.1 The basic ingredients: L, P, D, and U
100(2)
4.3.2 The main result
102(3)
4.3.3 Matrices with an LDU decomposition
105(2)
4.3.4 The Symmetric LDU Decomposition
107(1)
4.3.5 The Ranks of A and AT
108(5)
5 An Introduction to the Theory of Determinants
113(22)
5.1 An Introduction to the Determinant Function
114(5)
5.1.1 The main theorem
114(1)
5.1.2 The computation of a determinant
115(4)
5.2 The Definition of the Determinant
119(11)
5.2.1 The signature of a permutation
119(2)
5.2.2 The determinant via Leibniz's Formula
121(1)
5.2.3 Consequences of the definition
122(1)
5.2.4 The effect of row operations on the determinant
123(2)
5.2.5 The proof of the main theorem
125(1)
5.2.6 Determinants and LPDU
125(1)
5.2.7 A beautiful formula: Lewis Carroll's identity
126(4)
5.3 Appendix: Further Results on Determinants
130(5)
5.3.1 The Laplace expansion
130(2)
5.3.2 Cramer's Rule
132(2)
5.3.3 The inverse of a matrix over Z
134(1)
6 Vector Spaces
135(62)
6.1 The Definition of a Vector Space and Examples
136(5)
6.1.1 The vector space axioms
136(2)
6.1.2 Examples
138(3)
6.2 Subspaces and Spanning Sets
141(4)
6.2.1 Spanning sets
142(3)
6.3 Linear Independence and Bases
145(6)
6.3.1 The definition of linear independence
145(2)
6.3.2 The definition of a basis
147(4)
6.4 Bases and Dimension
151(11)
6.4.1 The definition of dimension
151(1)
6.4.2 Some examples
152(1)
6.4.3 The Dimension Theorem
153(3)
6.4.4 Finding a basis of the column space
156(1)
6.4.5 A Galois field application
157(5)
6.5 The Grassmann Intersection Formula
162(7)
6.5.1 Intersections and sums of subspaces
162(1)
6.5.2 Proof of the Grassmann intersection formula
163(2)
6.5.3 Direct sums of subspaces
165(2)
6.5.4 External direct sums
167(2)
6.6 Inner Product Spaces
169(14)
6.6.1 The definition of an inner product
169(1)
6.6.2 Orthogonality
170(3)
6.6.3 Hermitian inner products
173(1)
6.6.4 Orthonormal bases
174(1)
6.6.5 The existence of orthonormal bases
175(1)
6.6.6 Fourier coefficients
176(1)
6.6.7 The orthogonal complement of a subspace
177(1)
6.6.8 Hermitian inner product spaces
178(5)
6.7 Vector Space Quotients
183(4)
6.7.1 Cosets of a subspace
183(1)
6.7.2 The quotient V/W and the dimension formula
184(3)
6.8 Appendix: Linear Coding Theory
187(10)
6.8.1 The notion of a code
187(1)
6.8.2 Generating matrices
188(1)
6.8.3 Hamming distance
188(2)
6.8.4 Error-correcting codes
190(2)
6.8.5 Cosets and perfect codes
192(1)
6.8.6 The hat problem
193(4)
7 Linear Mappings
197(42)
7.1 Definitions and Examples
197(8)
7.1.1 Mappings
197(1)
7.1.2 The definition of a linear mapping
198(1)
7.1.3 Examples
198(2)
7.1.4 Matrix linear mappings
200(1)
7.1.5 An Application: rotations of the plane
201(4)
7.2 Theorems on Linear Mappings
205(6)
7.2.1 The kernel and image of a linear mapping
205(1)
7.2.2 The Rank-Nullity Theorem
206(1)
7.2.3 An existence theorem
206(1)
7.2.4 Vector space isomorphisms
207(4)
7.3 Isometries and Orthogonal Mappings
211(11)
7.3.1 Isometries and orthogonal linear mappings
211(1)
7.3.2 Orthogonal linear mappings on R"
212(1)
7.3.3 Projections
213(1)
7.3.4 Reflections
213(2)
7.3.5 Projections on a general subspace
215(1)
7.3.6 Dimension two and the O(2, R)-dichotomy
216(2)
7.3.7 The dihedral group as a subgroup of O(2, R)
218(1)
7.3.8 The finite subgroups of O(2, R)
219(3)
7.4 Coordinates with Respect to a Basis and Matrices of Linear Mappings
222(10)
7.4.1 Coordinates with respect to a basis
222(1)
7.4.2 The change of basis matrix
223(2)
7.4.3 The matrix of a linear mapping
225(1)
7.4.4 The Case V = W
226(2)
7.4.5 Similar matrices
228(1)
7.4.6 The matrix of a composition T o S
228(1)
7.4.7 The determinant of a linear mapping
228(4)
7.5 Further Results on Mappings
232(7)
7.5.1 The space L(V, W)
232(1)
7.5.2 The dual space
232(2)
7.5.3 Multilinear maps
234(1)
7.5.4 A characterization of the determinant
235(4)
8 Eigentheory
239(58)
8.1 The Eigenvalue Problem and the Characteristic Polynomial
239(12)
8.1.1 First considerations: the eigenvalue problem for matrices
240(1)
8.1.2 The characteristic polynomial
241(2)
8.1.3 The characteristic polynomial of a 2 × 2 matrix
243(1)
8.1.4 A general formula for the characteristic polynomial
244(7)
8.2 Basic Results on Eigentheory
251(8)
8.2.1 Eigenpairs for linear mappings
251(1)
8.2.2 Diagonalizable matrices
252(2)
8.2.3 A criterion for diagonaiizability
254(1)
8.2.4 The powers of a diagonalizable matrix
255(1)
8.2.5 The Fibonacci sequence as a dynamical system
256(3)
8.3 Two Characterizations of Diagonaiizability
259(9)
8.3.1 Diagonalization via eigenspace decomposition
259(2)
8.3.2 A test for diagonaiizability
261(7)
8.4 The Cayley--Hamilton Theorem
268(6)
8.4.1 Statement of the theorem
268(1)
8.4.2 The real and complex cases
268(1)
8.4.3 Nilpotent matrices
269(1)
8.4.4 A proof of the Cayley--Hamilton theorem
269(2)
8.4.5 The minimal polynomial of a linear mapping
271(3)
8.5 Self Adjoint Mappings and the Principal Axis Theorem
274(9)
8.5.1 The notion of self-adjointness
274(1)
8.5.2 Principal Axis Theorem for self-adjoint linear mappings
275(2)
8.5.3 Examples of self-adjoint linear mappings
277(1)
8.5.4 A projection formula for symmetric matrices
278(5)
8.6 The Group of Rotations of R3 and the Platonic Solids
283(11)
8.6.1 Rotations of R3
283(3)
8.6.2 The Platonic solids
286(1)
8.6.3 The rotation group of a Platonic solid
287(1)
8.6.4 The cube and the octahedron
288(2)
8.6.5 Symmetry groups
290(4)
8.7 An Appendix on Field Extensions
294(3)
9 Unitary Diagonalization and Quadratic Forms
297(22)
9.1 Schur Triangularization and the Normal Matrix Theorem
297(8)
9.1.1 Upper triangularization via the unitary group
298(1)
9.1.2 The normal matrix theorem
299(1)
9.1.3 The Principal axis theorem: the short proof
300(1)
9.1.4 Other examples of normal matrices
301(4)
9.2 Quadratic Forms
305(8)
9.2.1 Quadratic forms and congruence
305(1)
9.2.2 Diagonalization of quadratic forms
306(1)
9.2.3 Diagonalization in the real case
307(1)
9.2.4 Hermitian forms
308(1)
9.2.5 Positive definite matrices
308(2)
9.2.6 The positive semidefinite case
310(3)
9.3 Sylvester's Law of Inertia and Polar Decomposition
313(6)
9.3.1 The law of inertia
313(2)
9.3.2 The polar decomposition of a complex linear mapping
315(4)
10 The Structure Theory of Linear Mappings
319(18)
10.1 The Jordan--Chevalley Theorem
320(8)
10.1.1 The statement of the theorem
320(2)
10.1.2 The multiplicative Jordan--Chevalley decomposition
322(1)
10.1.3 The proof of the Jordan--Chevalley theorem
323(1)
10.1.4 An example
324(2)
10.1.5 The Lie bracket
326(2)
10.2 The Jordan Canonical Form
328(9)
10.2.1 Jordan blocks and string bases
328(1)
10.2.2 Jordan canonical form
329(1)
10.2.3 String bases and nilpotent endomorphisms
330(3)
10.2.4 Jordan canonical form and the minimal polynomial
333(1)
10.2.5 The conjugacy class of a nilpotent matrix
334(3)
11 Theorems on Group Theory
337(46)
11.1 Group Actions and the Orbit Stabilizer Theorem
338(11)
11.1.1 Group actions and G-sets
338(3)
11.1.2 The orbit stabilizer theorem
341(1)
11.1.3 Cauchy's theorem
341(2)
11.1.4 Conjugacy classes
343(1)
11.1.5 Remarks on the center
344(1)
11.1.6 A fixed-point theorem for p-groups
344(1)
11.1.7 Conjugacy classes in the symmetric group
345(4)
11.2 The Finite Subgroups of SO(3, R)
349(5)
11.2.1 The order of a finite subgroup of SO(3, R)
349(2)
11.2.2 The order of a stabilizer Gp
351(3)
11.3 The Sylow Theorems
354(5)
11.3.1 The first Sylow theorem
354(1)
11.3.2 The second Sylow theorem
355(1)
11.3.3 The third Sylow theorem
355(1)
11.3.4 Groups of order 12, 15, and 24
356(3)
11.4 The Structure of Finite Abelian Groups
359(5)
11.4.1 Direct products
359(2)
11.4.2 The structure theorem for finite abelian groups
361(1)
11.4.3 The Chinese Remainder Theorem
362(2)
11.5 Solvable Groups and Simple Groups
364(10)
11.5.1 The definition of a solvable group
364(2)
11.5.2 The commutator subgroup
366(1)
11.5.3 An example: A(5) is simple
367(2)
11.5.4 Simple groups and the Jordan-Holder theorem
369(1)
11.5.5 A few brief remarks on Galois theory
370(4)
11.6 Appendix: S(n), Cryptography, and the Enigma
374(5)
11.6.1 Substitution ciphers via S(26)
374(1)
11.6.2 The Enigma
375(2)
11.6.3 Rejewski's theorem on idempotents in S(n)
377(2)
11.7 Breaking the Enigma
379(4)
12 Linear Algebraic Groups: an Introduction
383(20)
12.1 Linear Algebraic Groups
383(15)
12.1.1 Reductive and semisimple groups
385(1)
12.1.2 The classical groups
386(1)
12.1.3 Algebraic tori
386(2)
12.1.4 The Weyl group
388(2)
12.1.5 Borel subgroups
390(1)
12.1.6 The conjugacy of Borel subgroups
391(1)
12.1.7 The flag variety of a linear algebraic group
392(1)
12.1.8 The Bruhat decomposition of GL(n, F)
393(2)
12.1.9 The Bruhat decomposition of a reductive group
395(1)
12.1.10 Parabolic subgroups
396(2)
12.2 Linearly reductive groups
398(5)
12.2.1 Invariant subspaces
398(1)
12.2.2 Maschke's theorem
398(1)
12.2.3 Reductive groups
399(1)
12.2.4 Invariant theory
400(3)
Bibliography 403(4)
Index 407
James B. Carrell is Professor Emeritus of mathematics at  the University of British Columbia. His research areas include algebraic transformation groups, algebraic geometry, and Lie theory.
Lisainfo e-raamatute kohta Lisainfo e-raamatute kohta
Püsilink: https://www.kriso.ee/db/97803877942806e.html
Märksõnad: