阿里云-飞天开放平台技术白皮书.pdf
阿里云计算为中国第一大公有云平台，云计算产品服务完全基于自主知识产权，先后获 85 项国家技术专利,获得国家发改委的云计算专项资金支持。阿里云飞天开放平台是在数据中心的大规模 Linux 集群之上构建的一套综合性的软硬件系统，将数以千计的服务器联成一台“超级计算机”，并且将这台超级计算机的存储资源和计算资源，以公共服务的方式，输送给互联网上的用户或者应用系统。阿里云致力于打造云计算的基础服务平台，注重为中小企业提供大规模、低成本的云计算服务。阿里云的目标是通过构建飞天这个支持多种不同业务类型的公有云计算平台，帮助中小企业在云服务上建立自己的网站和处理自己的业务流阿里云allyn. com打造数据分享第一平台飞天开放平台技术白皮书1.概述阿里云计算为中国第一大公有云平台,云计算产品服务完全基于自主知识产权,先后获85项国家技术专利,获得国家发改委的云计算专项资金支持。阿里云飞天开放平台是在数据中心的大规模 Linux集群之上构建的一套综合性的软硬件系统,将数以千讣的服务器联成一台“超级讣算机”,并且将这台超级计算机的存储资源和计算瓷源,以公共服务的方式,输送给互联网上的用户或者应川系统阿里云致力于打造云计算的基础服务平台,注重为中小企业提供大规模、低成本的云计算服务。阿里云的目标是通过构建飞天这个支持多种不同业务类型的公有云计算平台,帮助中小企业在云服务上建立自己的网站和处理自己的业务流程,帮助开发者向云端开发模式转变,用方便、低廉的方式让互联网服务全面融入人们的生活,将网络经济模式带入移动互联网,构建出以云计算为基础的全新互联网生态链。在此基础上,实现阿里云成为互联网数据分享第一平台的目标。2.体系架构如图2.1所示是飞天的体系架构图。整个飞天平台包括飞天内核(图2中黄色组件)和飞天开放服务(图2.1中蓝色组件)两大组成部分。飞天内核为上层的飞天开放服务提供存储、计算和调度等方面的底层支持,对应于图2.1中的协调服务、远程过程调用、安全管理、资源管理、分布式文件系统、任务调度、集群部署和集群监控模块飞天开放服务为用户应用程序提供了存储和计算两方面的接口和服务,包括弹性计算服务( Elastic Compute service,简称ECS)、开放存储服务(OpenStorage service,简称0SS)、开放结构化数据服务( Open table service,简称0TS)、关系型数据库服务( Relational database service,简称RDS)和开放数据处理服务( pen Data processing Service,简称ODPS),并基于弹性讣算服务提供了云服务引擎( Aliyun cloud engine,简称ACE)作为第三方应用阿里云allyn. com打造数据分享第一平台开发和Web应用运行和托管的平台。服务引擎ACE弹性计算关系型数开放存储开放构化开放数据据库服务数括服务处理服务RDSODPS分布式立科系统(盡白)任务调度(伏〕远程过程调用【夸父安全管理钟馗)分布协同服(女赐(伏)Linux集群数据飞天平台飞天内核飞大内核包含的模块可以分为以下儿部分:分布式系统底层服务:提供分布式环境下所需要的协调服务、远程过程调用、安全管理和资源管理的服务。这些底层服务为上层的分布式文件系统、任务调度等模块提供支持。分布式文件系统:提供一个海量的、可靠的、可扩展的数据存储服务,将集群中各个节点的存储能力聚集起来,并能够自动屏蔽软硬件故障,为用户提供不问断的数据访问服务。支持增量扩容和数据的自动平衡,提供类似于P0sSIX的用户空间文件访问API,攴持随机读写和追加写的操作仼务调度:为集群系统中的任务提供调度服务,同时支持强调响应速虔的在线服务( Online service)和强调处理数据吞吐量的离线任务( BalchProcessing job)。自动检测系统中故障和热点,通过错误重试、针对长尾作业并发备份作业等方式,保证作业稳定可靠地完成。集群监控和部署:对集群的状态和上层应川服务的运行状态和性能指标进行监控,对异常事件产生警报和记录;为运维人员提供整个飞天平台以及上层应用的部署和配置管理,支持在线集群扩容、缩容和应用服务的在线升级。阿里云allyn. com打造数据分享第一平台3.分布式系统底层服务31.协调服务(女娲)女妫(Nuwa)系统为飞天提供高可用的协调服务( Coordination service),是构建各类分布式应用的核心服务,它的作用是采用类似文件系统的树形命名空来让分布式进程互相协同工作。例如,当集群变更导致特定的服务被迫改变物理运行位置时,如服务器或者网终故障、配置调整或者扩容时,借助女可以使其他程序快速定位到该服务新的接入点,从而保证了整个平台的高可靠性和髙可用性。女娲基于类 Paxos协议[15],由多个女娲 Server以类似文件系统的树形结构存储数据,提供高可用、高并发用户请求的处理能力。女娲的目录表小一个包含文件的集合。与Unix中的文件路径一样,女娲中路径是以“/”分割的,根目录( Root entry)的名字是“/”,所有目录的名字都是以“/”结尾的。与Unix文件路径不一样之处在于:女娲中所有文件或日录都必须使用从根目录开始的绝对路径。由于女娲系统的设计目的是提供协调服务,而不是存储大量数据的,所以每个文件的内容( Value)的大小被限制在IMB以内。在女娲系统中,每个文件或目录都保存有创建者的信息。一旦某个路径被用户创建,其他用户就可以访问和修改这个路径的值(即文件内容或目录包含的文件名)。女娲攴持 Publish/ Subscribe模式,其中一个发布者、多个订阅者(OnePublisher/ Many Subscriber)的模式提供了基本的订阅功能:另外,还可用通过多个发布者、多个订阅者( Many Publisher/ Many Subscriber)的方式提供分布式选举( Distributed election)和分布式锁的功能。再举一个使用女娲来实现负载均衡的例子:提供某一服务的多个节点,在服务启动的时候在女娲系统的同一日录下创建文件,例如, server1创建文件nuwa:// cluster/ myservice/ server1”, server2在同一目录下创建“nuwa:// cluster/ myservice/ server2”。当客户端使用远程过程调用的时候」首先列举女娲服务中“nuwa:// cluster/ myservice”目录下的文件,这样就可以荻得 server1和 server2,客户端随后可以从中选择一个节点发出自己的请求阿里云allyn. com打造数据分享第一平台从而实现负载均衡。32.远程过程调用(夸父)在分布式系统中,不同计算机之间只能通过消息交换的方式进行通信。显式的消息通信必须通过 Socket接口编程,而远程过程调用( Remote procedureCall,简称RPC[9])可以隐藏显式的消息交换,使得程序员可以像调用本地函数一样来调用远程的服务夸父( Kuafu)是飞天内核中负责网络通信的模块,它提供了一个RPC的接,简化编写基于网络的分布式应用。夸父的设计目标是提供高可用(7x24小时)、大吞吐量( Gigabyte)、高效率、易用(简明APⅠ、多种协议和编程接口)的RPC服务。RPC客户端( RPC CLient)通过URI指定请求需要发送的RPC服务端( RPC Server)的地址,目前夸父支持两种协议形式。TCP:例如,tep:// fooserver01:9000●Ⅶuwa:例如,nuwa:/nuwa01/ Fooserver与用流( stream)传输的TCP通信相比,夸父通信是以消息( Message)为单位的,支持多种类型的消息对象,包括标准字符串std:: string和基于td:map实现的若干 string键值对。夸父RPC同时攴持异步( asynchronous)和同步( synchronous)的远程过程调用形式。异步调用:RPC函数调用吋不等接收到结果就会立即返回;用户必须通过显式调用接收函数取得请求结果。●同步调用:RPC函数调用时会等待,直到接收到结果才返回。在实现中,同步调用是通过封装异步调用来实现的。在夸父的实现中,客户端程序通过 Unix domain socket与本机上的一个夸父代理( Kuafu proxy)连接,不同计算机之间的夸父代理会建立一个TCP连接这样做的好处是可以更高效地使用网络带宽,系统可以支持上千台计算机之间的互联需求。此外,夸父利用女娲来实现负载均衡;对大块数据的传输做了优化与TCP类似,夸父代理之问还实现了发送端和接收端的流控( Flow Coηtrol)机制阿里云allyn. com打造数据分享第一平台33.安全管理(钟馗)钟馗( Zhongkui)是飞天内核中负责安全管理的模块,它提供了以用户为单位的身份认证和授权,以及对集群数据资源和服务进行的访问控制。用户的身份认证( Authentication)是于密钥机制的。用户对资源的访问控制是基于权能( Capability)机制进行授权( Authorization)的Capability是用于访问控制的一种数据结构,它定义∫对一个或多个指定的资源(如目录、文件、表等)所具有的访问权限。用户访问飞天系统的资源时必须持有 Capability,否则即视为非法。打个比方,如果把 Capability理解为地铁票,乘坐地铁(对地铁的一种访问方式)的时候必须要有 Capability,即地铁票。密钥对是基于公开密钥方法的,包括一个私钥和相对应的公钥。在飞天系统中,密钥对用于数字签名服务,以保证 Capability的不可伪造。换句话说,私钥用于生数字签名(如签发 Capability),公钥用于验证数字签名的有效性(如验证签发过的 Capability的有效性)考虑到网络通信时任何通信节点都是不可信的,所以即使是飞大自身模块内部之间的通信也同样是需要认证和授权的,而且验证的机制也完全一样。34.分布式文件系统(盘古)盘古( Pangu)是一个分布式文件系统,盘古的设计目标是将大量通用机器的存储资源聚合在一起,为用户提供大规模、高可靠、高可用、高吞吐量和冋扩展的存储服务,是飞天内核中的一个重要组成部分。大规模:能够支持数十PB量级的存储大小(1PB-1000T3),总文件数量达到亿量级。数据高可靠性:保证数捃和元数据( Metadata)是持久保存并能够正确访问的,保证所有数据存储在处于不同机架的多个节点上面(通常设置为3)。即使集群中的部分节点岀现硬件和软件故障,系统能够检测到故障并自动进行数据的备份和迁移,保证数据的纹全存在5阿里云allyn. com打造数据分享第一平台服务高可用性:保证用户能够不中断地访问数据,降低系统的不可服务时间。即使岀现软硬件的故障、异常和系统升级等情况,服务仍可正常访问。髙昋吐量:运行时系统Ⅰ/0吞吐量能够随机器规模线性增长,保证响应时间高可扩展性:保证系统的容量能够通过增加机器的方式得到白动扩展,卜线札器存储的数据能够自动迁移到新加入的节点上同时,盘古也能很好地支持在线应用的低延时需求。在盘古系统中,文件系统的元数据存储在多个主服务器( Master)上,文件内容存储在人量的块服务器( Chunk server)上。客户端程序在使用盘古系统时,首先从主服务器获取元数据信息(包括接下来与哪些块服务器交互),然后在块服务器上直接进行数据操作。由于元数据信息很小,大量的数据交互是客户端直接与块服务器进行的,因此盘占采用少量的主服务器来管理元数据,并使用 Paxos协议[15]保证元数据的致性。此外,块大小被设置为64MB,进一步减少了元数据的大小,因此可以将元数据全部放到内存里,从而使得主服务器能够处理大量的并发请求块服务器负责存储大小为64B的数据块。在向文件写入数据之前,客户端将建立到3个块服务器的连接,客户向主副本( Replica)写入数据以后,由主副本负责向其他副本发送数据。与直接由客户端向三个副本写入数据相比,这样可以减少客户端的网终带宽使用。块副本在放置的时候,为保证数据可用性和最大化地使用网络带宽,会将副本放置在不同机架上,并优先考虑磁盘利用率低的杋器。当硬件故障或数据不可用造成数据块的副本数目达不到3份的时候,数据块会被重新复制。为保证数据的完整性,每块数据在写入时会同时计算一个校验值,与数据同时写入磁盘。当读取数据块的时候,块服务器会再次计算恔验值与之前存入的值是否相同,如果不同就说明数据出现了错误,需要从其他副木重新读取数据。在线应用对盘古提出了与离线应用不同的挑战:OSS、OTS要求低吋延数据读写,ECS在要求低吋延的同吋还需要具备随机写的能力。针对这些需求,盘古实现了事务日志文件和随机访问文件,用以支撑在线应用。其中,日志文件通过阿里云allyn. com打造数据分享第一平台多种方法对时延进行了优化,包括设置更高的优先级、由客户端直接写多份拷贝而不是用传统的流水线方式、写入成功,不经过 Master确认等。随机访问文件则允许用户随机读写,同时也应用了类似日志文件的时延优化技术。35.资源管理和任务调度(伏羲)伏羲(Fuxi)是飞天内核中负责资源管理和任务调度的模块,同时也为应用开发提供了一套编程基础框架。伏羲同时支持强调响应速度的在线服务和强调处理数据吞吐量的离线任务。在伏羲中,这两类应用分别简称为 Service和Job在资源管理方面,伏羲主要负责调度和分配集群的存储、计算等资源给上层应用;管理运行在集群节点上任务的生命周期;在多用户运行环境中,支持讣算额度、访问控制、作业优先级和资源抢占,达到在保障公平的前提下,有效地共享集群资源。在任务调度方面,伏羲囿向海量数据处理和大规模计算类型的复杂应用,提供了一个数据驱动的多级流水线并行计算框架,在表述能力上兼容MapReduce l12」、Map- Reduce-erge等多种编程模式;自动检测故障和系统热点,重试失败任务,保证作业稳定可靠运行完成;具有高可扩展性,能够根据数据分布优化网终开销。伏羲中应用了“ Master/ Worker”工作模型。其中, Master负责进行资源中请和调度、为 Worker创建工作计划(Plan)并监控 Worker的生命周期, Worker负责执行具体的工作计划并及时向 Master汇报工作状态( Status)。此外, Master支持多级模式,即一个 Master可以隶属于另外一个 Master之下伏羲 Master负责整个集样资源管理和调度,处理Job/ Service启动、停止Failover等生命周期的维护。同时伏羲 Master支持多用户额度配冒、Job/ Service的多优先级设置和动态资源抢占逻辑,可以说是飞天的“大脑”伏羲对资源调度是多维度的,可以根据CPU、内存等系统资源,以及应用自定义的虚拟资源对整个机群进行资源分配和调度土伯(Tubo)是部署在每台由伏羲管理的机器上的后台进程,负责收集并向伏羲 Master报告本机的状态,包括系统资源的消耗、 Master或 Worker进程的运行、等待、完成和失败事件,并根据伏羲 Master或者Job/ Service master阿里云allyn. com打造数据分享第一平台的指令,启动或杀死指定的 Master或 Worker进程。同时土伯还负责对计算机健康状况进行监控,对异常 Worker(比如内存超用)进行及时的清理和汇报对于在线服务( Service),由伏羲 Master负责 Service master的启动与状态监控,处理相应 Service master的资源申请请求。 Service master负责管理Service Worker的任务分配、生命周期管理以及 Failover的管理。对于离线任务(Job),伏羲 Master负责 Job Master的启动与状态监控,处理相应 Job master的资源申请凊求。 Job master根据用户输入的Job描述文件,将仼务分解成一个或以上的Task,每个Task的资源申请、 Task Worker的调度和生命周期维护由 Task master负责3.5.1.在线服务调度在飞天内核中,每个 Service都有一个 Service master和多个不同角色(Role)的 Service worker,它们一起协同工作来完成整个服务的功能。 ServiceMaster是伏羲 Master管理下的子 Master( Child master),它负责这个 Service相关的资源申请、状态维扩以及故障恢复,并定期与伏羲 Master进行交互,确保整个 Service正确、正常地运行。每个 Service Worker的角色和执行的动作,都是由用户来定义的每个 Service Worker负责处理一个到多个数据分片( Partition),同一时刻一个分片只会被分型到一个 Service Worker处理。将数据分割成为互不相关的分片,然后将不同分片给不同 Service worker来处理是构建大规模应川服务的关键特性。数据分片是一个抽象的概念,在不同的应川中有不同的含义。在服务运行的过程中,每个 Seryice的数据分片的数和内容都是可以动态变化的,应用程序可以根据实际需要对数据分片动态地进行加载(Load)、卸载( Unload)、分裂( Split)和迁移( Migrate)等操作。3.5.2.离线任务调度在飞天中,一个离线任务(Job)的执行过程被抽象为一个有向无环图( Directed Acyclic graph,简称DAG):图上每个顶点对应一个Task,每条边对应一个 Pipeline。个连接的两个Task的 Pipeline表示前个Task的输出
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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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