Exercises in Algebra : A Collection of Exercises in Algebra, Linear Algebra, and Geometry
Book information
Description
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
Similar books
MySQL® Notes for Professionals book
2018 · PDF
MrExcel 2022: Boosting Excel
2022 · PDF
MrExcel 2022: Boosting Excel
2022 · PDF
Session C11: Ancient Cultural Landscapes in South Europe – their Ecological Setting and Evolution, Session C22: Gardeners from South America, Session S04: Agro-Pastoralism and Early Metallurgy Sessions, Session WS29: The Idea of Enclosure in Recent Iberian Prehistory, Session C88: Rhytmes et causalites des dynamiques de l'anthropisation en Europe entre 6500 ET 500 BC: Hypotheses socio-culturelles et/ou climatiques: Proceedings of the XV UISPP World Congress (Lisbon 4-9 September 2006) / Actes du XV Congrès Mondial (Lisbonne 4-9 Septembre 2006) Vol.36
2010 · PDF
THE BRITISH ARMY IN INDIA: ITS PRESERVATION BY AN APPROPRIATE CLOTHING, HOUSING, LOCATING, RECREATIVE EMPLOYMENT, AND HOPEFUL ENCOURAGEMENT OF THE TROOPS. with AN APPENDIX ON INDIA : THE CLIMATE OP ITS HILLS ; THE DEVELOPMENT OF ITS RESODRCBS, INDUSTRY, AND ARTS ; THE ADMINISTRATION OF JUSTICE ; THE BLACK ACT ; THE PROGRESS OF CHRISTIANITY ; THE TRAFFIC IN OPIUM ; THE VALUE OF INDIA ; PERMANENT CAUSES OF DISAFFECTION, AND OF THE RECENT REBELLION ; THE TRADITIONARY POLICY; MISGOVERNMENT BY NATIVE RULERS ; ANNEXATIONS OF THEIR TERRITORY, ETC.
1858 · PDF
Idries Shah 27 Books Collection : A Perfumed Scorpion, A Veiled Gazelle, Caravan of Dreams, Darkest England, Destination Mecca, Evenings with Idries Shah, Knowing How to Know, Learning How to Learn, Letters and Lectures of Idries Shah, Neglected aspects of Sufi study, Observations, Oriental Magic, Reflections, Seeker after Truth, Special Illumination, Special Problems in the study of Sufi ideas, Sufi thought and action, Tales of the Dervishes, The Dermis Probe, The Elephant in the Dark, The Englishman Handbook, Idries Shah Antology, The Magic Monastery, The natives are restless, wisdom of the Idiots PDF.
2022 · PDF
The travels of Capts. Lewis and Clarke from St. Louis, by way of the Missouri and Columbia rivers, to the Pacific ocean; performed in the years 1804, 1805 & 1806, by order of the government of the United States. Containing delineations of the manners, customs, religion, &c. of the Indians, comp. from various authentic sources, and original documents, and a summary of the Statistical view of the Indian nations, from the official communication of Meriwether Lewis. Illustrated with a map of the country, inhabited by the western tribes of Indians
1809 · PDF
Professional Linux kernel architecture ''Wrox programmer to programmer''--Cover. - ''What you are reading right now is the result of an evolution over more than seven years: After two years of writing, the first edition was published in German by Carl Hanser Verlag in 2003. It then described kernel 2.6.0. The test was used as a basis for the low-level design documentation for the EAL4+ security evaluation of Red Hat Enterprise Linux 5, requiring to update it to kernel 2.6.18 (if the EAL acronym does not mean anything to you, then Wikipedia is once more your friend). Hewlett-Packard sponsored the translation into English and has, thankfully, granted the rights to publish the result. Updates to kernel 2.6.24 were then performed specifically for this book''--P. ix
2008 · PDF