开关磁阻电机系统理论与控制技术
开关磁阻电机系统理论与控制技术，介绍了基本理论，电机设计，Matlab仿真内容提要本书共分为8章分别阐述了开关磁阻电机及其控制系统发展概况,推导了电机线性、准线性和非线性数学模型,给出了开关磁阻电机计算设计程序,讲述了开关磁阻电机有限元分析方法,研究了开关磁阻电机调速系统的控制策略,详细介绍了利用软件建立开关磁阻电机仿真模型的步骤,井进行了稳态性能仿真和动态性能仿真,最后针对DSP对开关磁阻电机有位置传感器和无位置传感器调速系统进行理论分析与设计开本书适用于从事电力电子及电气传动专业高等学院教师和研究生,以及相关专业好、运的科研机构的研究人员和直流关磁阻木空把矿、航功率范本磁阻电图书在版编目(CIP)数据上的拓机线性开关磁阻电机系统理论与控制技术/吴红星编著.一北京:中国了开关电力出版社,20107略,以(现代工业自动化技术应用丛书)电SBN978-7-5123-0336-2行有限:1.①开…Ⅱ.①吴…Ⅲ.①开关控制-磁阻电机-系统理在Max论②开关控制-磁阻电机控制Ⅳ.①TM352线:对车中国版本图书馆CIP数据核字(2010)第070773号系统仿J结构来》电枝该模型E控制器矩分配巨中国电力出版社出版、发行机的发E北京三里河路6号1004htp/www.cspp.com.cn)磁阻电积北京丰源印剧厂印刷电右各地新华书店经售建立开弓2010年8月第一版2010年8月北京第一次印剧统稳态忙787毫米X1092毫米16开本17印张452千字了优化。印数000-3000册定价36.00元电机调这敬告读者法,推毛木书封面贴有防伪标签,加热后中心图案消失各模型过本书如有印装质量问题,我社发行部负责退换电初版权专有翻印必究关磁阻欠压保折前厂言开关磁阻电机调速系统具有结构简单、坚固、工作可靠、成本低、系統控制灵活、调速性能好、运行效率高、温升低等诸多优点,它综合了交流变频调速系统的坚固耐用、适用于恶劣环境和直流调速系统的可控性好等优良特性,被专家视为电气传动系统发展过程中的一个里程碑。开关磁阻电机特别适用在恶劣环境和要求超高速的场合下运行,并可广泛地应用在纺织、造纸、煤矿、航空、机槭等领域的造纸机、浆纱机、采煤机,风机、水泵、家用电器和机器人等负载上,功率范围从几瓦到儿兆瓦,转速范围从几转到儿万转。本书的宗旨是,着眼于实用技术,并兼顾到发展趋势。考虑到实际应用的需要,介绍了开关磁阻电机的几种结构形式,针对新型开关磁阻电机进行论述,总结开关磁阻电机在绕组结构形式上的拓扑结构,论述各种绕组拓扑结构的优缺点。在开关磁阻电机基本方程式的基础上,推导电机线性数学模型和准线性数学模型,具体分析绕组电流、绕组磁链、绕组电感和电磁转矩,给出了开关磁阻电机设计步骤,并分析了转矩脉动产生的原因,研究开关磁阻电机调速系统的控制策略,以DSP为控制芯片,给出了开关磁阻电机调速系统设计方法和基本设计软件。电机本体设计方面:给出计算程序,对128电机进行各类参数的计算:对开关磁阻电机进行有限元分析:利用 Ansoft软件建立开关磁阻电机的有限元模,用 RMxprt得到二维几何模型,在 Maxwell2D的瞬态求模块下进行有限元分析;分析得到的绕组电流、绕组磁链、电磁转矩曲线:对转子极弧系数、轴径、开通角等参数进行优化分析:在分析有限元计算的矩角特性曲线和系统仿真后的转矩输出波形的基础上,得出产生转矩脉动的根本原因,通过改进电机定子磁极的结构来减小气隙磁场的突变,通过修改气隙等参数从而减小和抑制转矩脉动。电机控制策略方面:根据数学模型研究基于永磁磁通控制开关磁阻电机非线性数学模型。在该模型的基础上,研究基于永磁磁通控制开关磁阻电机调速系统的控制策略。设计绕组电流闭环控制器、转速调节控制器,硏究基于永磁磁通控制开关磁阻电机转矩分配的控制策略,推导了转矩分配函数,并设计转矩控制器。分析开关磁阻电机的发电运行机理和能流关系,对开关磁阻电机的发电运行理论进行线性分析,推导基本电路方程和相电流解析式;通过线性模型,分析开关磁阻电机的有效发电条件电机仿真技术方面:用 MATLAB软件建立开关磁阻电机的准非线性动态仿真模型的基础上,建立开关磁阻电机系统的系统模型,并对系统模型进行了稳态性能仿真和动态性能仿真。利用系统稳态性能仿真,综合考虑最大平均转矩和效率这两个优化目标,对升关磁阻电机的开关角进行了优化。针对传统PⅠ控制策略对开关醚阻电机调速系统进行仿真,得到采用传统P控制策略的电机调速系统的电机相电流波形和系统转矩波形。深入研究基于模糊控制的控制理论和控制方法,推导基于模糊控制的搾制算法,提出一种模糊PI相结合的控制方法,并建立仿真模型,对各模型进行比较,以便得到最佳控制策略电机控制系统方面:介绍开关磁阻电机调速特点,分析电机驱动功率电路拓扑结构,介绍开关磁阻电机调速系统转子位置传感器分类及使用方法。设计了驱动电路,过流保护电路、过压和欠压保护电路、电机专用控制电路等硬件。利用T公司的电机专用DsP设计开关磁阻电机有位全书共8章,第1章介绍了开关磁阻电机调速系统的概况、发展趋势及主要应用领域。第2章介绍了开关磁阻电机的线性数学模型、准线性数学模型及非线性数学模型。第3章分析了开关磁阻电机的各类损耗,介绍了开关磁阻电机本体的设计方法。第4章利用 Ansoft软件建立开关磁阻电机的有限元模,用 RMxprt得到二维几何模型,在 Maxwell2D的瞬态模块下进行有限元分析。分析得到的绕组电流、绕组磁链、电磁转矩曲线。对转子极弧系数、轴径、开通角等参数进行优化分析。第5章介绍了开关磁阻电机调速系统在各类调速系统的地位,设计开关磁阻调速控制系统硬件。第6章针对开关磁阻电机调速特性研究开关磁阻电机控制策略和发电机理。第7章用 MATLAB软件建立开关磁阻电机系統的系统模型,并对系统模型进行稳态性能仿真和动态性能仿真。第8章针对DSP对开关磁阻电机有位置传感器和无位置传感器调速系统进行理论分析与设计。前言本书由吴红星编著,各章编写工作有赵晢、嵇恒、刘莹、钱海荣、黄冬林、倪天、郭庆波、第1章叶宇骄等参与。全书由吴红星统稿,寇宝泉教授支持本书的编写并审阅了书稿。编写过程中,参1.1阅和利用了国内外大量文献、资料,在此对原作者一并致谢。1.2限于作者水平,加上时间仓促,缺点、错误在所难免,热忱欢迎广大读者批评指正。1,2,1,2,21,3于14J14.114.214.314.414.514.614.714.81.5开1.6开第2章2.1开22开2.2,122.32,242,2.52.3开24开2,4.124.22.4.324424.5主要应用领域。第2第3章分析了开关soft软件建立开关模块下进行有限元径、开通角等参数最》设计开关磁阻调速略和发电机理。第7态性能仿真和动态速系统进行理论分前言第1章绪论、倪天、郭庆波、11开关磁阻电机的发展概况…编写过程中,参12开关磁阻电机的结构特点…,+…日2…2121开关磁阻电机的优点………者批评指正。1.2.2开关磁阻电机的缺点和和国国面自和“““““国目目围把的和一41.3开关磁阻电机的优化方法141.4开关磁阻电机系统抑制转矩脉动技术…1.4.1基于抑制转矩脉动的传统控制策略…14.2基于抑制转矩脉动的线性化控制…614.3基于抑制转矩脉动的变结构控制61.44基于抑制转矩脉动的智能控制理论…14.5基于抑制转矩脉动的转矩分配策略…………146基于抑制转矩脉动的迭代学习控制…1014.7基于抑制转矩脉动的微步控制策略…………101.4.8其他方法015开关磁阻电机未来研究方向…1.6开关磁阻电机的工业应用………………………2第2章开关磁阻电机的工作原理及数学模型142.1开关磁阻电机基本原理22开关磁阻电机的一些基本结构…1422.1单相开关磁阻电机……………142.22两相开关磁阻电机……………152.2.3三相开关磁阻电机………………52.24四相开关磁阻电机……………………………………1622.5五相以上开关磁阻电机1623开关磁阻电机改进结构…1624开关磁阻电机数学模型……………21224.1电路方程…2.42机械方程2224,3机电联系方程*…,…222.44线性模型……1232,4.5准线性模型25混合励磁开关磁阻电机数学模型…日型道日副上理福2.5混合励磁电机磁路特点………………252混合励磁开关磁阻电机转矩平衡方程……4446i4.4.7孟第3章开关磁阻电机电磁设计……………………3744863.1开关磁阻电机设计及优化方法……………13945有限31.1电机本体结构设计………94.5.R…………393.1.2电机参数优化设计………4.52M32开关磁阻电机损耗分析……45.3有321绕组铜损分析…………42046基于车3.2.2机械损耗分析……404.6.1影3.2.3杂散损耗分析…44.62开…………41324电机铁损分析……………………………………………………414.6.3定33开关磁阻电机参数计算…+45第5章开关331电负荷与磁负荷…51开关磁332主要尺寸4534开关磁阻电机本体设计示例…………475.12电34.1相数,极数和绕组端电压……………5.3电压34.2主要尺寸的选择计计4852开关磁343其他结构尺寸及绕组匝数”5.3开关磁阝48344电流及转矩计算……………………………………5053.1开关34.5绕组设计……………………………………………15053.2开关34.6参数计算533开关51534开关第4章开关磁阻电机性能优化…54开关磁阵41电机电磁场的理论基础………………5454.l标准42有限元法54542其他421有限元法的发展55…555.5迭代学买42.2 Ansoft软件简介……………………………555.1基于423 Ansoft有限元法…55,2迭代424电磁场有限元方法的特点及一般步骤65.6开关磁43 RMxprt软件设计及使用方法………15756.1速度43.1启动软件……………5858562转矩432新建SRM模型563电流4.3.3建模结果……68第6章开关磁434仿真计算61开关磁阳43.5模型导出…1111736.1.1与步44 Maxwell2D软件设计及使用方法…………“环“746.12与反44.1打开工程文件………m…“+t74613与直442模型设置……17614与无44.3材料设置………786.1.5与异444边界及激励源设置806.2功率电子44.6设置仿真参数+,90……34……*34447运动部分设置……………………………………………9544.8仿真运算96+:37…014.5有限元分析结果处理………39451 RMxprt输出的性能曲线………10139452 Maxwell2D的求解结果…394.3有限元后处理………1044046基于转矩波动抑制电机本体优化…1-11054046.1影响转矩波动的因素…404.62开通角、关断角对转矩波动的影响…………………106…41463定子磁极结构对转矩波动的影响……………xx+107414第5章开关磁阻电机的控制策略…5.开关磁阻电机控制方式……,中日副日是…:1045…:4551l角度位置控制(APC)1111045512电流斩波控制(CCC)……………………………1l475.1.3电压斩波控制(CVC)……………4852开关磁阻电机调速特性53开关磁阻电机能量回馈控制…::1:1111248…4853.1开关磁阻电机发电运行机理…50532开关磁阻电机发电运行的励磁过程:50533开关磁阻电机的能量变换理论……………………………1451534开关磁阻电机发电状态工作特点……………”…11654开关磁阻电机PD控制::18*54541标准数字PID算法………………19:54542其他PD方法1120……5555迭代学习控制…121:5551基于模型控制系统和迭代学习控制系统概述…………:121555.52迭代学习控制过程和开环PD迭代学习控制……122“565.6开关磁阻电机的转矩分配控制系统设计…123……*5756.!速度调节器设计……………123585.6.2转矩分配函数的设计………425+58563电流控制器设计……………1265968第6章开关磁阻电机调速系统硬件设计……………128696,1开关磁阻电机调速系统在电机控制中的地位1128…736.L.1与步进电动机驱动系统的比较+++:128…746.1.2与反应式同步电动机的比较…………28746.1.3与直流电动机的比较“…自129a776.14与无换向器直流电动机的比较:1297861.5与异步电动机变频调速系统的比较………1298062功率电子器件…………………130622功率GBT工作特点……1317463PWM控制技术…11137.4.6.3.1传统PWM技术…计计……1347.463.2优化后的PWM技术“国计1347.4633空间电压矢量PWM控制…计………13574.6.34跟踪型PWM控制技术1357464开关磁阻电机控制器功率拓扑结构………-13674641不对称半桥主回路…137第8章642H桥主回路…积+…137643不对称半桥改进型”…1378.2644(n+1)型功率变换器1388264.5电容裂相型…“……1388,264.6电容转储型…+1…:1388365整流及吸收回路设计………1398.36.5.1功率吸收电路设计…………………139836.52吸收电路参数计算…::+140846.53整流电路设计……846.54电流采样与处理电路…道上中中和国和8,46.5.5转子位置信号采集与处理……………………1428.5656系统保护电路设计:1498.66.6功率及驱动电路11518.66.61SKH24驱动模块在SRD系统中的应用…………518.6.62S19976DY—桥式驱动器的原理及应用11528.76.63EXB841工作原理…578.8664FCAS50SN60开关磁阻电机功率模块…………1598.第7章基于DSP开关磁阻电机控制器设计……1658.7.DSP的特点1657.2电动机DSP控制系统基础……………………167参考文721DSP电机控制特点…1677.22数字滤波DSP实现方法…………16873有位置传感器DSP控制++117173.1开关磁阻电机控制机理……“已,171732DSP控制开关磁阻电机硬件设计…………………………17733软件设计x+2:::175734电流控制………………………17773.5位置控制178736速度控制……国+…“……“…“…面,,正重自7.3.7换相控制………………181738速度控制器……………x1837.3.9DSP编程示例………………………8474开关磁阻电机无传感器DSP控制…………………:21674.1调速系统硬件描述………………216…133742无传感器开关磁阻电机驱动系统的控制软件…………216…:134743无传感器换相和速度更新算法………………………………218134744速度环:21135745电流控制回路……213574.6斜坡控制器……136747无传感器开关磁阻电机驱动系统的校准………………2313第8章开关磁阻电机调速系统仿真……22813781引言+*13782基于 MATLAB/Simulink的系统建模与仿真分析……128138821仿真软件 MATLAB/Simulink简介…x:12813882,2电机模型的建立……………………23…13883控制系统P控制策略建模与仿真……23013983.1SRM调速系统的无P控制仿真………30…13983.2电机调速系统的PI控制仿真分析……………31…,……,:14084基于模糊控制器的系统仿真分析……………236…:148.4.1模糊控制器的设计1236142842SRM调速系统的模糊控制仿真及结果分析…………………23814285sRM调速系统模糊P控制仿真……………………242………1498.6开关磁阻电机能量回馈建模与仿真…………24351…………151861发电状态的基本电路方程…86.2发电运行的相电流解析…24315287开关磁阻电机控制系统模型分析……2455788开关磁阻电机发电系统模型的建立….…:2521598.81电流滞环控制模块………………………254,+1658.82电流计算模块………2541658.8.3转矩计算模块…254167参考文献……::::::1256…………16716817117117317577178179181中“183184216
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Finite-Dimensional Vector Spaces - P. Halmos (Springer, 1987)
在学习代数学之余，值得一看的代数学书籍。里面介绍了更为丰富的代数学概念和结论。PREFACEMy purpose in this book is to treat linear transformations on finite-dimensional vector spaces by the methods of more general theories. Theidea is to emphasize the simple geometric notions common to many partsof mathematics and its applications, and to do so in a language that givesaway the trade secrets and tells the student what is in the back of the mindsof people proving theorems about integral equations and Hilbert spaces.The reader does not, however, have to share my prejudiced motivationExcept for an occasional reference to undergraduate mathematics the bookis self-contained and may be read by anyone who is trying to get a feelingfor the linear problems usually discussed in courses on matrix theory orhigher"algebra. The algebraic, coordinate-free methods do not lose powerand elegance by specialization to a finite number of dimensions, and theyare, in my belief, as elementary as the classical coordinatized treatmentI originally intended this book to contain a theorem if and only if aninfinite-dimensional generalization of it already exists, The temptingeasiness of some essentially finite-dimensional notions and results washowever, irresistible, and in the final result my initial intentions are justbarely visible. They are most clearly seen in the emphasis, throughout, ongeneralizable methods instead of sharpest possible results. The reader maysometimes see some obvious way of shortening the proofs i give In suchcases the chances are that the infinite-dimensional analogue of the shorterproof is either much longer or else non-existent.A preliminary edition of the book (Annals of Mathematics Studies,Number 7, first published by the Princeton University Press in 1942)hasbeen circulating for several years. In addition to some minor changes instyle and in order, the difference between the preceding version and thisone is that the latter contains the following new material:(1) a brief dis-cussion of fields, and, in the treatment of vector spaces with inner productsspecial attention to the real case.(2)a definition of determinants ininvariant terms, via the theory of multilinear forms. 3 ExercisesThe exercises(well over three hundred of them) constitute the mostsignificant addition; I hope that they will be found useful by both studentPREFACEand teacher. There are two things about them the reader should knowFirst, if an exercise is neither imperative "prove that.., )nor interrogtive("is it true that...?" )but merely declarative, then it is intendedas a challenge. For such exercises the reader is asked to discover if theassertion is true or false, prove it if true and construct a counterexample iffalse, and, most important of all, discuss such alterations of hypothesis andconclusion as will make the true ones false and the false ones true. Secondthe exercises, whatever their grammatical form, are not always placed 8oas to make their very position a hint to their solution. Frequently exer-cises are stated as soon as the statement makes sense, quite a bit beforemachinery for a quick solution has been developed. A reader who tries(even unsuccessfully) to solve such a"misplaced"exercise is likely to ap-preciate and to understand the subsequent developments much better forhis attempt. Having in mind possible future editions of the book, I askthe reader to let me know about errors in the exercises, and to suggest im-provements and additions. (Needless to say, the same goes for the text.)None of the theorems and only very few of the exercises are my discovery;most of them are known to most working mathematicians, and have beenknown for a long time. Although i do not give a detailed list of my sources,I am nevertheless deeply aware of my indebtedness to the books and papersfrom which I learned and to the friends and strangers who, before andafter the publication of the first version, gave me much valuable encourage-ment and criticism. Iam particularly grateful to three men: J. L. Dooband arlen Brown, who read the entire manuscript of the first and thesecond version, respectively, and made many useful suggestions, andJohn von Neumann, who was one of the originators of the modern spiritand methods that I have tried to present and whose teaching was theinspiration for this bookP、R.HCONTENTS的 FAPTERPAGRI SPACESI. Fields, 1; 2. Vector spaces, 3; 3. Examples, 4;4. Comments, 55. Linear dependence, 7; 6. Linear combinations. 9: 7. Bases, 108. Dimension, 13; 9. Isomorphism, 14; 10. Subspaces, 16; 11. Calculus of subspaces, 17; 12. Dimension of a subspace, 18; 13. Dualspaces, 20; 14. Brackets, 21; 15. Dual bases, 23; 16. Reflexivity, 24;17. Annihilators, 26; 18. Direct sums, 28: 19. Dimension of a directsum, 30; 20. Dual of a direct sum, 31; 21. Qguotient spaces, 33;22. Dimension of a quotient space, 34; 23. Bilinear forms, 3524. Tensor products, 38; 25. Product bases, 40 26. Permutations41; 27. Cycles,44; 28. Parity, 46; 29. Multilinear forms, 4830. Alternating formB, 50; 31. Alternating forms of maximal degree,52II. TRANSFORMATIONS32. Linear transformations, 55; 33. Transformations as vectors, 5634. Products, 58; 35. Polynomials, 59 36. Inverses, 61; 37. Mat-rices, 64; 38. Matrices of transformations, 67; 39. Invariance,7l;40. Reducibility, 72 41. Projections, 73 42. Combinations of pro-jections, 74; 43. Projections and invariance, 76; 44. Adjoints, 78;45. Adjoints of projections, 80; 46. Change of basis, 82 47. Similarity, 84; 48. Quotient transformations, 87; 49. Range and null-space, 88; 50. Rank and nullity, 90; 51. Transformations of rankone, 92 52. Tensor products of transformations, 95; 53. Determinants, 98 54. Proper values, 102; 55. Multiplicity, 104; 56. Triangular form, 106; 57. Nilpotence, 109; 58. Jordan form. 112III ORTHOGONALITY11859. Inner products, 118; 60. Complex inner products, 120; 61. Innerproduct spaces, 121; 62 Orthogonality, 122; 63. Completeness, 124;64. Schwarz e inequality, 125; 65. Complete orthonormal sets, 127;CONTENTS66. Projection theorem, 129; 67. Linear functionals, 130; 68. P aren, gBCHAPTERtheses versus brackets, 13169. Natural isomorphisms, 138;70. Self-adjoint transformations, 135: 71. Polarization, 13872. Positive transformations, 139; 73. Isometries, 142; 74. Changeof orthonormal basis, 144; 75. Perpendicular projections, 14676. Combinations of perpendicular projections, 148; 77. Com-plexification, 150; 78. Characterization of spectra, 158; 79. Spec-ptral theorem, 155; 80. normal transformations, 159; 81. Orthogonaltransformations, 162; 82. Functions of transformations, 16583. Polar decomposition, 169; 84. Commutativity, 171; 85. Self-adjoint transformations of rank one, 172IV. ANALYSIS....17586. Convergence of vectors, 175; 87. Norm, 176; 88. Expressions forthe norm, 178; 89. bounds of a self-adjoint transformation, 17990. Minimax principle, 181; 91. Convergence of linear transformations, 182 92. Ergodic theorem, 184 98. Power series, 186APPENDIX. HILBERT SPACERECOMMENDED READING, 195INDEX OF TERMS, 197INDEX OF SYMBOLS, 200CHAPTER ISPACES§L. FieldsIn what follows we shall have occasion to use various classes of numbers(such as the class of all real numbers or the class of all complex numbers)Because we should not at this early stage commit ourselves to any specificclass, we shall adopt the dodge of referring to numbers as scalars. Thereader will not lose anything essential if he consistently interprets scalarsas real numbers or as complex numbers in the examples that we shallstudy both classes will occur. To be specific(and also in order to operateat the proper level of generality) we proceed to list all the general factsabout scalars that we shall need to assume(A)To every pair, a and B, of scalars there corresponds a scalar a+called the sum of a and B, in such a way that(1) addition is commutative,a+β=β+a,(2)addition is associative, a+(8+y)=(a+B)+y(3 there exists a unique scalar o(called zero)such that a+0= a forevery scalar a, and(4)to every scalar a there corresponds a unique scalar -a such that十(0(B)To every pair, a and B, of scalars there corresponds a scalar aBcalled the product of a and B, in such a way that(1)multiplication is commutative, aB pa(2)multiplication is associative, a(Br)=(aB)Y,( )there exists a unique non-zero scalar 1 (called one)such that al afor every scalar a, and(4)to every non-zero scalar a there corresponds a unique scalar a-1or-such that aaSPACES(C)Multiplication is distributive with respect to addition, a(a+n)If addition and multiplication are defined within some set of objectsscalars) so that the conditions(A),B), and (c)are satisfied, then thatset(together with the given operations) is called a field. Thus, for examplethe set Q of all rational numbers(with the ordinary definitions of sumand product)is a field, and the same is true of the set of all real numberaand the set e of all complex numbersHHXERCISIS1. Almost all the laws of elementary arithmetic are consequences of the axiomsdefining a field. Prove, in particular, that if 5 is field and if a, and y belongto 5. then the following relations hold80+a=ab )Ifa+B=a+r, then p=yca+(B-a)=B (Here B-a=B+(a)(d)a0=0 c=0.(For clarity or emphasis we sometimes use the dot to indi-cate multiplication.()(-a)(-p)(g).If aB=0, then either a=0 or B=0(or both).2.(a)Is the set of all positive integers a field? (In familiar systems, such as theintegers, we shall almost always use the ordinary operations of addition and multi-lication. On the rare occasions when we depart from this convention, we shallgive ample warningAs for "positive, "by that word we mean, here and elsewherein this book, "greater than or equal to zero If 0 is to be excluded, we shall say"strictly positive(b)What about the set of all integers?(c) Can the answers to these questiong be changed by re-defining addition ormultiplication (or both)?3. Let m be an integer, m2 2, and let Zm be the set of all positive integers lessthan m, zm=10, 1, .. m-1). If a and B are in Zmy let a +p be the leastpositive remainder obtained by dividing the(ordinary) sum of a and B by m, andproduct of a and B by m.(Example: if m= 12, then 3+11=2 and 3. 11=9)a) Prove that i is a field if and only if m is a prime.(b What is -1 in Z5?(c) What is囊izr?4. The example of Z, (where p is a prime)shows that not quite all the laws ofelementary arithmetic hold in fields; in Z2, for instance, 1 +1 =0. Prove thatif is a field, then either the result of repeatedly adding 1 to itself is always dif-ferent from 0, or else the first time that it is equal to0 occurs when the numberof summands is a prime. (The characteristic of the field s is defined to be 0 in thefirst case and the crucial prime in the second)SEC. 2VECTOR SPACES35. Let Q(v2)be the set of all real numbers of the form a+Bv2, wherea and B are rational.(a)Ie(√2) a field?(b )What if a and B are required to be integer?6.(a)Does the set of all polynomials with integer coefficients form a feld?(b)What if the coeficients are allowed to be real numbers?7: Let g be the set of all(ordered) pairs(a, b)of real numbers(a) If addition and multiplication are defined by(a月)+(,6)=(a+y,B+6)and(a,B)(Y,8)=(ary,B6),does s become a field?(b )If addition and multiplication are defined by(α,月)+⑦,b)=(a+%,B+6)daB)(,b)=(ay-6a6+的y),is g a field then?(c)What happens (in both the preceding cases)if we consider ordered pairs ofcomplex numbers instead?§2. Vector spaceWe come now to the basic concept of this book. For the definitionthat follows we assume that we are given a particular field s; the scalarsto be used are to be elements of gDEFINITION. A vector space is a set o of elements called vectors satisfyingthe following axiomsQ (A)To every pair, a and g, of vectors in u there corresponds vectora t y, called the aum of a and y, in such a way that(1)& ddition is commutative,x十y=y十a(2)addition is associative, t+(y+2)=(+y)+a(3)there exists in V a unique vector 0(called the origin) such thata t0=s for every vector and(4)to every vector r in U there corresponds a unique vector -rthat c+(-x)=o(B)To every pair, a and E, where a is a scalar and a is a vector in u,there corresponds a vector at in 0, called the product of a and a, in sucha way that(1)multiplication by scalars is associative, a(Bx)=aB)=, and(2 lz a s for every vector xSPACESSFC B(C)(1)Multiplication by scalars is distributive with respect to vectorddition, a(+y=a+ ag, and2)multiplication by vectors is distributive with respect to scalar ad-dition, (a B )r s ac+ Bc.These axioms are not claimed to be logically independent; they aremerely a convenient characterization of the objects we wish to study. Therelation between a vector space V and the underlying field s is usuallydescribed by saying that v is a vector space over 5. If S is the field Rof real number, u is called a real vector space; similarly if s is Q or if gise, we speak of rational vector spaces or complex vector space§3. ExamplesBefore discussing the implications of the axioms, we give some examplesWe shall refer to these examples over and over again, and we shall use thenotation established here throughout the rest of our work.(1) Let e(= e)be the set of all complex numbers; if we interpretr+y and az as ordinary complex numerical addition and multiplicatione becomes a complex vector space2)Let o be the set of all polynomials, with complex coeficients, in avariable t. To make into a complex vector space, we interpret vectoraddition and scalar multiplication as the ordinary addition of two poly-nomials and the multiplication of a polynomial by a complex numberthe origin in o is the polynomial identically zeroExample(1)is too simple and example (2)is too complicated to betypical of the main contents of this book. We give now another exampleof complex vector spaces which(as we shall see later)is general enough forall our purposes.3)Let en,n= 1, 2,. be the set of all n-tuples of complex numbers.Ix=(1,…,轨)andy=(m1,…,n) are elements of e, we write,,bdefinitionz+y=〔1+叽,…十物m)0=(0,…,0),-inIt is easy to verify that all parts of our axioms(a),(B), and (C),52, aresatisfied, so that en is a complex vector space; it will be called n-dimenaionalcomplex coordinate space
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