Exercises in Algebra : A Collection of Exercises in Algebra, Linear Algebra, and Geometry

Front Matter
	Cover
	Half Title
	Title Page
	Copyright Page
	Table of Contents
Preface to the Russian edition
Foreword to the English edition
PART ONE: FOUNDATIONS OF ALGEBRA
	1: Sets and maps
		1. Operations on subsets. Calculation of the number of elements
			Q 101-102
			Q 103-106
			A 103-105
		2. Calculation of the number of maps and of the number of subsets. Binomial coefficients
			Q 201-203
			Q 204-211
			Q 212-213
			A 202-213
		3. Permutations
			Q 301
			A 301
			Q 302-305
			A 302-306
			Q 306-308
			Q 309-317
			Q 318-323
			A 307-320
			A 321-323
		4. Recurrence relations. Induction
			Q 401-406
			Q 407-411
			Q 412-414
			A 402-414
		5. Sununations
			Q 501-503
			Q 504-506
			A 501-502
			A 503-506
	2: Arithmetic spaces and linear equations
		6. Arithmetic spaces
			Q 601-603
			Q 604-608
			Q 609-612
			A 601-612
			Q 613-618
			A 613-618
		7. Rank of a matrix
			Q 701
			Q 702-709
			Q 710-715
			Q 716-719
			A 701-719
		8. Systems of linear equations
			Q 801
			A 801
			Q 802
			Q 803
			A 802-803
			Q 804
			Q 806-812
			A 804-810
			Q 813-820
			Q 821-823
			Q 824-826
			A 811-826
	3: Determinants
		9. Determinants of sizes two and three
			Q 901-902
			A 901-902
		10. Expanding a determinant. Inductive definition
			Q 1001-1004
			Q 1005-1007
			A 1001-1006
		11. Basic properties of a determinant
			Q 1101
			Q 1102-1109
			A 1101-1105
			Q 1110
			A 1106-1110
		12. Expanding a determinant according to the elements of a row or a column
			Q 1201
			Q 1202-1203
			Q 1204
			A 1201-1204
		13. Calculating a determinant with the help of elementary operations
			Q 1301
			A 1301
			Q 1302-1303
			A 1302-1303
		14. Calculating special determinants
			Q 1401
			A 1401
		15. Determinant of a product of matrices
			Q 1501-1503
			Q 1504
			A 1501-1504
		16. Additional exercises
			Q 1601-1605
			A 1601
			A 1602-1613
			Q 1606-1611
			Q 1612-1615
			Q 1616-1620
			A 1614-1620
			Q 1621
			A 1621
	4: Matrices
		17. Operations on matrices
			Q 1701
			Q 1702-1705
			A 1701-1704
			Q 1706-1710
			Q 1711-1725
			A 1705-1726
			Q 1726-1729
			A 1727-1729
		18. Matrix equations. Inverse matrix
			Q 1801-1803
			Q 1804
			Q 1805-1809
			A 1801-1808
			A 1809-1810
			Q 1810-1815
			Q 1816-1824
			A 1811-1824
		19. Special matrices
			Q 1901-1902
			Q 1903-1907
			A 1902-1909
			A 1912-1927
			Q 1908-1927
	5: Complex numbers
		20. Complex numbers in algebraic form
			Q 2001-2004
			Q 2005-2011
			A 2001-2005
			A 2008-2011
		21. Complex numbers in trigonomebic form
			Q 2101-2103
			Q 2104-2111
			A 2101
			A 2102-2111
			Q 2112-2114
			A 2113-2114
		22. Roots of complex numbers. Cyclotomie polynomials
			Q 2201
			Q 2203-2207
			A 2206-2207
			Q 2208-2214
			Q 2215-2221
			A 2208-2220
			Q 2222-2227
			A 2222-2225
			A 2226-2227
		23. Calculation of sums and products with the help of complex numbers
			Q 2301
			Q 2302
			Q 2303-2307
			A 2301-2305
			A 2307
		24. Complex numbers and geometry of a plane
			Q 2401-2406
			A 2402-2406
			Q 2407-2415
			Q 2416-2424
			A 2407-2424
			Q 2425-2426
			A 2426
	6: Polynomials
		25. Division with a remainder. Euclidean algorithm
			Q 2501
			Q 2503-2508
			A 2501-2508
			Q 2509
		26. Simple and multiple roots over fields of characteristic zero
			Q 2601-2602
			Q 2603-2610
			Q 2611-2614
			A 2601-2614
		27. Prime decomposition over R and C
			Q 2701
			Q 2702-2711
			A 2701-2708
			Q 2712-2714
			A 2709-2714
		28. Polynomials over the field of rationals and over finite fields
			Q 2801-2806
			Q 2807-2817
			A 2801-2809
			Q 2818-2832
			A 2810-2826
			Q 2833-2834
			A 2827-2834
		29. Rational fractions
			Q 2901-2902
			A 2901-2902
			Q 2903-2905
			A 2903-2905
		30. Interpolation
			Q 3001-3004
			Q 3005-3014
			A 3001-3012
		31. Symmetric polynomials. Vieta formulas
			Q 3101-3109
			A 3101
			A 3102-3110
			Q 3110-3111
			Q 3112-3120
			Q 3121-3127
			Q 3128-3131
			Q 3132-3134
			A 3112-3133
		32. Resultant and discriminant
			Q 3201-3203
			A 3201
			Q 3204-3212
			A 3202-3212
			Q 3213-3217
			A 3213-3217
		33. Isolation of roots
			Q 3301-3303
			Q 3304-3312
			A 3301-3310
			Q 3313-3320
			A 3312-3320
PART TWO: LINEAR ALGEBRA AND GEOMETRY
	7: Vector spaces
		34. Concept of a vector space. Bases
			Q 3401-3402
			Q 3403-3407
			A 3402-3407
			Q 3408-3410
			Q 3411-3414
			A 3408-3413
		35. Subspaces
			Q 3501
			Q 3502
			Q 3503-3506
			A 3501-3503
			Q 3507-3511
			Q 3512-3515
			A 3504-3515
			Q 3516-3520
			Q 3521-3527
			A 3516-3524
		36. Linear hmetions and mappings
			Q 3601
			Q 3602-3609
			A 3601-3610
			Q 3610-3617
			Q 3618-3621
			A 3611-3621
	8: Bilinear and quadratic functions
		37. General bilinear and sesquilinear functions
			Q 3701
			Q 3702-3705
			A 3701-3705
			Q 3706-3710
			Q 3711-3714
			Q 3715-3718
			Q 3719-3721
			Q 3722-3729
			A 3706-3726
			Q 3730-3733
			Q 3734-3744
			Q 3745-3746
			A 3728-3745
		38. Synunetric bilinear, Hermitian and quadratic functions
			Q 3801
			Q 3802-3807
			Q 3808-3812
			A 3801-3808
			Q 3813-3818
			Q 3819-3821
			Q 3822-3831
			Q 3832-3835
			A 3809-3832
	9: Linear operators
		39. Definition of a linear operator. The image, the kernel, the matrix of a linear operator
			Q 3901-3904
			Q 3905-3915
			A 3901-3915
			Q 3916-3918
			Q 3919-3923
			Q 3924
			A 3917-3924
		40. Eigenvectors, invariant subspaces, root subspaces
			Q 4001-4006
			A 4001-4006
			Q 4007-4015
			Q 4016-4024
			A 4008-4018
			Q 4025-4032
			Q 4042-4044
			A 4019-4043
		41. Jordan canonical form and its applications. Minimal polynomial
			Q 4101
			A 4101
			Q 4102-4105
			Q 4106-4117
			A 4103-4106
			Q 4118-4126
			A 4107-4126
			Q 4127-4135
			Q 4136-4144
			Q 4145-4146
			A 4127-4147
		42. Normed vector spaces and algebras. Non-negative matrices
			Q 4201-4202
			Q 4203-4209
			Q 4210-4218
			A 4202-4213
			Q 4219-4229
			Q 4230-4234
			A 4214-4234
	10: Metric vector spaces
		43. Geometry of metric spaces
			Q 4302-4307
			Q 4308-4315
			Q 4316-4319
			A 4304-4319
			Q 4320-4322
			Q 4323-4328
			Q 4329-4336
			Q 4337-4344
			A 4321-4344
			A 4345
			Q 4345-4349
		44. Adjoint and normal operators
			Q 4401
			Q 4402-4409
			Q 4410-4415
			Q 4416-4424
			A 4402-4423
			Q 4425-4432
			A 4424-4427
		45. Selfadjoint operators. Reduction of quadratic functions to principal axes
			Q 4501-4505
			A 4504
			Q 4506-4514
			Q 4515-4520
			A 4507-4519
			Q 4521
		46. Orthogonal and unitary operators. Polar factorization
			Q 4601-4606
			A 4604-4606
			Q 4607-4609
			Q 4611-4618
			A 4607-4612
			Q 4619-4628
			Q 4629-4633
			A 4613-4632
	11: Tensors
		47. Basic concepts
			Q 4701-4703
			A 4701-4703
			Q 4704-4708
			Q 4709-4714
			Q 4716-4719
			A 4704-4719
		48. Symmetric and skew-symmetric tensors
			Q 4801-4802
			Q 4803-4809
			A 4802-4805
			Q 4810-4818
			A 4807-4815
	12: Affine, Euclidean, and projective geometry
		49. Affine spaces
			Q 4901-4908
			Q 4909-4915
			Q 4916-4920
			Q 4921-4927
			A 4909-4924
			Q 4928-4933
			Q 4934-4937
			A 4928-4937
			Q 4938
		50. Convex sets
			Q 5001-5006
			Q 5007-5013
			A 5003-5008
			Q 5018-5023
			A 5015-5022
			Q 5030-5034
			Q 5035-5037
			A 5023-5035
			A 5037
		51. Euclidean spaces
			Q 5101-5102
			Q 5103-5107
			Q 5108-5112
			Q 5113-5116
			Q 5117-5121
			A 5101-5117
			Q 5122-5126
			A 5118-5126
		52. Hypersuifaces of second degree
			Q 5201-5204
			A 5202-5204
			Q 5205-5208
			Q 5209-5216
			A 5205-5216
			Q 5217-5221
			A 5217-5221
			A 5222
			Q 5222
			Q 5223-5227
			Q 5228
			A 5223-5228
		53. Projective spaces
			Q 5301-5308
			A 5301-5304
			Q 5309-5322
			Q 5323-5329
			Q 5330-5334
			A 5305-5334
PART THREE: BASIC ALGEBRAIC STRUCTURES
	13: Groups
		54. Algebraic operations. Semigroups
			Q 5401-5404
			A 5401-5403
			Q 5405-5409
			A 5404-5406
		55. Concept of a group. Isomorphisms of groups
			Q 5501
			Q 5502-5505
			Q 5506
			Q 5507-5515
			Q 5516-5526
			Q 5527-5533
			A 5501-5531
			A 5533
		56. Subgroups. Orders of elements of groups. Cosets
			Q 5601-5603
			Q 5604-5608
			Q 5609-5616
			A 5601-5617
			Q 5617-5625
			Q 5626-5632
			A 5618-5630
			Q 5633-5634
			Q 5635-5640
			A 5631-5638
		57. Action of a group on a set. Relation of conjugacy
			Q 5701-5702
			Q 5703-5708
			A 5701-5705
			Q 5709-5714
			A 5709-5715
			Q 5715-5721
			Q 5722-5727
			A 5717-5727
			A 5728-5734
			Q 5728-5736
			Q 5737-5745
			A 5735-5745
		58. Homomorphisms and normal subgroups. Factorgroups and centers
			Q 5801
			Q 5802-5811
			A 5801-5803
			Q 5812-5818
			Q 5820-5828
			Q 5829-5840
			A 5804-5817
			A 5818-5832
			Q 5841-5848
			A 5833-5841
			A 5842-5848
		59. Sylow subgroups. Groups of small orders
			Q 5901-5902
			A 5901-5906
			Q 5903-5913
			A 5907-5913
			Q 5914-5925
			A 5914-5923
			Q 5926-5927
			A 5924-5926
		60. Direct products and direct sums. Abelian groups
			Q 6001-6011
			Q 6012-6021
			A 6001-6013
			Q 6022-6037
			A 6014-6028
			A 6029-6040
			Q 6038-6047
			Q 6048-6052
			A 6041-6055
			Q 6053-6066
			Q 6067-6069
			A 6056-6069
		61. Generators and defining relations
			Q 6101-6104
			A 6101
			Q 6105-6115
			A 6102-6115
			Q 6116-6127
			A 6116-6128
			Q 6128-6133
			Q 6134-6136
			A 6129-6136
		62. Solvable groups
			Q 6201-6208
			A 6201
			A 6202-6216
			Q 6209-6218
			Q 6219-6226
			Q 6227-6229
			A 6217-6227
			A 6228-6229
	14: Rings
		63. Rings and algebras
			Q 6301
			Q 6302-6303
			Q 6304-6311
			A 6301-6312
			Q 6312-6316
			Q 6317-6322
			Q 6323-6325
			Q 6326-6327
			A 6313-6327
			Q 6328-6330
			A 6330
		64. Ideals, homomorphisms, factor-rings
			Q 6401-6406
			Q 6407-6418
			A 6401-6417
			Q 6419-6426
			Q 6427-6430
			Q 6431-6437
			A 6418-6437
			Q 6438-6447
			Q 6448-6456
			A 6439-6456
			Q 6457-6465
			Q 6466-6474
			A 6457-6475
			Q 6475-6486
			Q 6487-6488
			A 6476-6486
		65. Special classes of algebras
			Q 6501-6503
			A 6501-6503
			Q 6504-6505
			Q 6506
			Q 6507-6511
			A 6504-6508
			Q 6512-6515
			Q 6516-6518
			A 6509-6515
		66. Fields
			Q 6601-6604
			Q 6605-6620
			A 6602-6612
			Q 6621-6632
			Q 6633-6637
			Q 6638-6644
			Q 6646-6648
			A 6614-6647
		67. Extensions of fields. Galois theory
			Q 6701-6702
			A 6701
			Q 6703-6711
			Q 6712-6717
			A 6706-6713
			Q 6718-6722
			Q 6723-6727
			Q 6728-6732
			A 6714-6732
			Q 6733-6740
			Q 6741-6749
			Q 6750-6755
			A 6733-6754
			Q 6756-6767
			A 6755-6769
			Q 6768-6779
			A 6770-6779
			Q 6780-6781
			A 6780-6781
		68. Finite fields
			Q 6801-6804
			Q 6805-6809
			Q 6810-6815
			A 6801-6812
			A 6813-6815
	15: Elements of representation theory
		69. Representations of groups. Basic concepts
			Q 6901-6904
			Q 6905-6911
			Q 6912-6916
			Q 6917-6922
			Q 6923-6929
			Q 6930-6931
			A 6904-6927
		70. Representations of finite groups
			Q 7001-7003
			Q 7004-7010
			A 7002-7005
			Q 7011-7024
			Q 7025-7034
			Q 7035-7048
			Q 7049-7052
			A 7006-7049
		71. Group algebras and their modules
			Q 7101-7104
			Q 7105-7116
			Q 7117-7127
			A 7102-7127
			Q 7128-7139
			Q 7140-7147
			Q 7148-7150
			A 7128-7145
		72. Characters of representations
			Q 7201-7210
			A 7201-7209
			Q 7211-7219
			Q 7220-7230
			A 7220-7226
			A 7227-7228
			A 7229-7233
			Q 7231-7234
			Q 7235-7243
			A 7234-7243
			Q 7244
		73. Initial information on representations of continuous groups
			Q 7301-7302
			Q 7303-7310
			A 7302-7305
			Q 7311-7313
			A 7306-7314
Answers and hints
Theoretical material
	1. Affine and Euclidean geometry
	2. Hypersurfaces of the second order
	3. Projective space
	4. Tensors
	5. Elements of representation theory
List of definitions
List of symbols
Index