UDS诊断程序，整车网络测试应用程序（PCAN-UDS API – User Manual.pdf）
UDS_PCAN_APIA应用程序，整车网络诊断应用程序，超值！（PEAK CAN UDS Application Programming InterfaceUser Manual.pdf）PCAN-UDS APi- User ManualContents1 PCAN-UDS API Documentation2 Introduction2.1 Understanding PCAN-UDS2.2 Using PCAN-UDS2.3 Features7888992.4 System Requi rements2.5 Scope of supply3 DLL API Reference3.1 Namespaces103.1.1 Peak Can uds3.2 Units3.21 PuDs Unit3.3 Classes3.3.1 UDSApi3.3.2 TUDSApi3. 4 structures1022334553.4.1 TPUDSMsg3.4.2 TPUDSSessionInfo3.43 TPUDSNetAddrinfo3.5 Types213.5.1 TPUDSCANHand]e223.5.2 TPUDSstatus233.5.3 TPUDSBaudrate253.5.4 TPUDSHWType283.5.5 TPUDSResult303.5.6 TPUDSParameter313.5.7 TPUDSService393.5.8 TPUDSAddress423.5.9 TPUDSCanId443.5.10 TPUDSProtoco l463.5.11 TPUDSAddressingType483.5.12 TPUDSMessageType493.5.13 TPUDSSVCParamDSC503.5.14 TPUDSSVCParamER513.5.15 TPUDSSVCParamcc533.5.16 TPUDSSVCParamTP543.5.17 TPUDSSVCParamcdTCS543.5.18 TPUDSSvCParamROE553.5.19 TPUDSSvCParamROERe commendedserviceID573.5.20 TPUDSSVCParamLC583.5.21 TPUDSSvcParamLCBaudrateidentifier593.5.22 TPUDSSVCParamDI603.5.23 TPUDSSVCParamRDBPI643.5.24 TPUDSSVCParamDDDI653,525 TPUDSSyCParamRDTCI66PCAN-UDS APi- User Manual3.5.26 TPUDSSVCParamRDTCI DTCSVM6935.27 TPUDSSYCParamIOCBI703.5.28 TPUDSSvCParamRC3.5.29 TPUDSSVCParaMRC RID723.6 Methods733.6.1 Initialize753.6.2 Initialize(TpudsCanhandle, tpudsbaudrate)3.6.3 Initialize(TPUdsCANhandle, TPUdSBaudrate, TPudSHWType, UInt32,UInt16)83.6.4 Uninitialize813.6.5 Setvalue843.6.6 Setvalue (TPUdsCanhandle, tpudsparameter, UInt32, uint32)843.6Setvalue (TPUdSCaNHandle, TPUDSParameter, stringBufferUint32)873.6.8 Setvalue (TPUDSANHandle, TPUDSParameter, Byte[], Uint32)883.6.9 Setvalue(Tpudscanhand le, tpudsparameter, IntPtr, UInt32)3.6.10 Getvalue933.6.11 Getvalue (TPUDSCANHandle, TPUDSParameter, StringBufferUint32)933.6. 12 Getvalue (TPUDSCANHandle, tpudsparameter, uint32, Uint32)963.6.13 Getvalue (TPUDsCaNHandle, TPUDSParameter, Byte l], UInt32)993.6. 14 Getvalue (TPUdSCAnhandle, tpudSParameter, Intptr, UInt32)1013.6.15 Getstatus1043.6.16Read1073.6.17 Write3.6.18 Reset1143.6.19 WaitForsing lemessage1163. 6.20 WaitFormultiplemessage1203.6.21 Waitforseryice1263.6.22 WaitForservicefunctional1303.6.23 ProcessResponse1333.6. 24 SvCDiagnosticsessioncontro l1383.6.25 SVCECUReset1413.6.26 SvcSecuri tyAccess1453.6.27 SvCCommunicationControl1483.6.28 SvcTesterpresent1523.6.29 SvcsecuredDataTransmission1553.6.30 SvcControlDTCSetting1583.6.31 SvcResponseonEvent1623.6.32 SVCLinkcontrol1663.6.33 SVCReaddatabyidentifier1703.6. 34 SvcReadMemory ByAddress1733.6.35 SvcReadscal ingdatabyidentifier1773.6. 36 SvcReadDataByperiodicIdentifier1803.6.37 SvcDynamicallydefinedataIdentifierDBID1843.6.38 SvcDynamicall ydefineDataIdentifierDBMA1883.6. 39 SvcDynamical lyDefineDataIdentifierCDDDI1933.6.40 SvcWri teDataByidentifier1973.6. 41 Svcwri teMemory byaddress2003.6.42 SvcClearDi agnosticInformation2053. 6. 43 SVCReadDTCInformation2083.6.44 SvCReadDTCInformationRDTCSSBDTC2113. 6. 45 SvCReaddTCInformationRDTCSSBRN215PCAN-UDS APi- User Manual3. 6.46 SVcReadDTCInformationReportExtended2183.6. 47 SvcReadDTCInformationReportseverity2213,648 SvcReaddTCInformationrsIodtc2253. 6.49 SvCReadDTCInformationNoParam2283.6.50 SvcInputout put contro byidentifier2323. 6.51 SyCRoutineControl2363.6.52 SvCReques tOwn load2393.6.53 SvcRequestUp load2433. 6.54 SVCTransferData2483.6.55 SvCRequestTransferExit2513.7 Functions2563.7.1 UDS Initialize2583.7.2 UDs Uninitialize2593.7.3 UDs Setvalue2603.7.4 UDs Getvalue2613.7.5 UDS Getstatus2623.7.6 UDS Read2643.7.7 UDs Write2653.7.8 UDs Reset2663.7.9 UDS_WaitForsinglemessage2673.7.10 UDS_waitForMultipleMessage2693.7.11 UDs Wai ce2723.7.12 UDS WaitForserviceFunctional2733.7.13 UDS_ Processresponse2753.7.14 UDS_SvcDiagnosticSessionControl2773.7.15 UDS SVCECUReset2783.7.16 DS_SVCSecuri tyAccess2803.7.17 UDS SVCCommunicationcontrol2813.7.18 UDs SvCTesterpresent2833719 UDS SvCSecuredDatatransmission2843.7.20 UDS_SvCControlDTCSetting2863.7.21 UDS_SVCResponseonEvent2873,7.22 UDs SVCLinkcontrol2893.7.23 UDS_SvcReaddatabyidentifier2913.7.24 UDS_SvcReadMemory byAddress2923.7.25 uDs_ SvcReadscalingdatabyidentifier2943.7.26 UDS_SvCReadDataBy Periodi iDentifier2953.7. 27 UDS_SVcDynamical l yDefineDataIdentifierDBID2973.7.28 UDS_SvcDynami call ydefinedataIdentifierDBMa2993.7.29 UDS_SvcDynami cal l yDefineDataIdentifierCDDDI3013. 7.30 UDS_SvcWriteDataByIdentifier3023,7.31 UDs SvcWri teMemorybyaddress3033.7. UDS_SvcClearDiagnosticInformation3053.7.33 UDS SVCReadDTCInformation3073.7. UDs SyCReadDTCInformationRdtCSSBDTC3093.7.35 uDs SvCReadDTCInformationRdtcssbrn3103.7.36 UDS_ SvCReadDTCInformationReportExtended3113.7.37 UDS_SvcReadDTCInformationReportseverity3133.7.38 UDS SVCReadDTCInformationRSIODTC3153,739 UDS SVCReadDTCInformationNoParam3163. 7.40 UDS_SvcInputoutput contro l byIdentifier3,7. 41 UDs SyCRoutinecontrol319PCAN-UDS APi- User Manual3.7.42 UDS_SvcRequestDown load3213.7.43 UDS_ SVCRequestupload32337.44 UDS SyCTransferData3253.7.45 UDS_SVCRequestTransferExit3263.8 Definitions3293.8.1 PCAN-UDS Handle Definitions3293.8.2 Parameter value defintions3313.8.3 TPUDSMsg Member value Definitions3323.8.4 PCAN-UDs Service parameter Definitions3334 Additional Information3354.1 PCAn Fundamentals33542 PCAN-Basic3364.3 UDS and ISO-TP Network Addressing Information3384.3.1 ISO-TP network addressing format3384.4 USing Events3405 License Information3426PCAN-UDS APi- User Manual1 PCAn-UDS APi DocumentationWelcome to the documentation of PCan-UD APl, a PEAK CAN API that implements ISo 15765-3, UDS in CANan international standard that allows a diagnostic tester(client) to control diagnostic functions in an on-vehicleElectronic Control Unit(ECU or serveIn the following chapters you will find all the information needed to take advantage of this aPlIntroduction on page 8DLL API Reference on page 10Additional Information on page 335PCAN-UDS APi- User Manual2 IntroductionPCAN-UDS is a simple programming interface intended to support windows automotive applications that usePEAK-Hardware to communicate with Electronic Control Units(ECU) connected to the bus systems of a car, formaintenance purpose2.1 Understanding PCAN-UDSUDS stands for Unified Diagnostic Services and is a communication protocol of the automotive industry. thisprotocol is described in the norm iSo 14229-1The UDS protocol is the result of 3 other standardized diagnostic communication protocolsIS0 14230-3, as known as Keyword 2000 Protocol(KWP2000L IS0 15765-3, as known as diagnostic on CANISo 15765-2, as known as ISo-TPThe idea of this protocol is to contact all electronic data units installed andCAN OBDninterconnected in a car, in order to provide maintenance, as checking for errors,actualizing of firmware, etcUDS is a Client/Server oriented protocol. In a UDS session(diagnostic session ),aprogram application on a computer constitutes the client(within UDS, it is calledPCAN-UDSTester), the server is the ecu being tested and the diagnostic requests from client toserver are called services. The client always starts with a request and this ends with apositive or negative response from the server(ECuSince the transport protocol of UDS is done using ISo-TP, an international standardPCAN ISOTPfor sending data packets over a CAN Bus, the maximum data length that can betransmitted in a single data-block is 4095 bytes.PCAN-UDS API is an implementation of the Uds on CAN standard the physicalcommunication is carried out by PCAN-Hardware (PCAN-USB, PCAN-PCI etc )throughPCAN-Basithe pCAN-ISo-TP and PCAN-Basic API (free CAN APls from PEAK-System). Because ofthis it is necessary to have also the pCAN-1S0-tP and PCAN-Basic APls(PCAN-ISO-TP. dll and PCAN Basic. dll) present on the working computer where UdS is intended tobe used. PCAN-UDS, PCAN-ISO-TP and PCan-Basic apis are free and available for allFigure 1: Relationship of thepeople that acquire a pCAn-hardware2.2 Using PCAN-UDSSince PCAN-UDS API is built on top of the PCAN-1So-TP API and PCAN-Basic APls, it shares similar functions. Itoffers the possibility to use several PCAn-UDS (PUds) channels within the same application in an easy way. Thecommunication process is divided in 3 phases: initialization interaction and finalization of a puds-channelInitialization In order to do UDS on CAN communication using a channel, it is necessary to initialize it first. Thisis done by making a call to the function UDS_ Initialize (class- method: InitializePCAN-UDS APi- User ManualInteraction: After a successful initialization a channel is ready to communicate with the connected can bus.Further configuration is not needed the 24 functions starting with UDS Svc(class-methods: starting with Svccan be used to transmit UdS requests and the utility functions starting with Uds WaitFor(class- methodsstarting with WaitFor) are used to retrieve the results of a previous request. the Uds read and UDS Write(class-methods: Read and Write are lower level functions to read and write UDs messages from scratch. Ifdesired, extra configuration can be made to improve a communication session, like service request timeouts orISo-TP parametersFinalization: When the communication is finished, the function UDS_ Uninitialize(class-method: Uninitializeshould be called in order to release the puds-channel and the resources allocated for it. In this way thechannel is marked as free"and can be used from other applications23 FeaturesI mplementation of the UDS protocol(iSo 14229-1)for the communication with control unitsWindows DLLs for the development of 32-bit and 64-bit applicationsPhysical communication via Can using a Can interface of the pcan seriesUses the pcan-Basic programming interface to access the can hardware in the computerUses the pCAn-ISo-TP programming interface(iso 15765-2)for the transfer of data packages up to 4095bytes via the can bus2.4 System Requi rementsL- Windows 10, 8.1, 7(32/64-bitAt least 512 Mb ram and 1 GHz CPUPC CAN interface from peak-SystemPCAN-Basic APlL PCAN-SO-TP API2.5 Scope of supplyInterface DLL, examples, and header files for all common programming languagesDocumentation in pdf formatDocumentation in HTML Help formatPCAN-UDS APi- User Manual3 DLL API ReferenceThis section contains information about the data types (classes, structures, types, defines enumerations)andAPI functions which are contained in the pcan-uds api3.1 NamespacesPEAK offers the implementation of some specific programming interfaces as namespaces for the. NEtFramework programming environment. The following namespaces are available:NamespacesNameDescription}PeakContains all namespaces that are part of the managed programming environment fromPEAK-SystemPeak CanContains types and classes for using the PCan aPi from PEAK-SystemPeak Can. LightContains types and classes for using the PCAn-Light API from PEAK-SystemPeak Can basicContains types and classes for using the pcan-Basic APl from PEAK-SystemPeak Can CcpContains types and classes for using the CCP API implementation from PEAK-SystemPeak Can XcpContains types and classes for using the XcP aPi implementation from PEAK-SystemPeak Can. Iso TpContains types and classes for using the pCAN-IS0-TP aPl implementation from PEAKSystelPeak Can, UdsContains types and classes for using the PCan-UDS API implementation from PEAK-SystemPeakCan.Obdll Contains types and classes for using the PCAN-OBDIll API implementation from PEAKSystemt}Peak. LinContains types and classes used to handle with lin devices from PEAK-Systemt}Peak. RP1210AContains types and classes used to handle with can devices from PEak-System through theTMC Recommended Practices 1210, version A, as known as RP1210(A3.1.1 Peak Can UdsThe peak Can. Uds namespace contains types and classes to use the pcan-UdS aPi within the. NET Frameworkprogramming environment and handle pcan devices from peak-SystemRemarks: Under the delphi environment, these elements are enclosed in the puds-Unit. the functionality of allelements included here is just the same. the difference between this namespace and the delphi unit consists inthe fact that delphi accesses the Windows api directly it is not managed code)AliasesAliasDescriptionTPUDSCANHandle Represents a pCAn-UDS channel handleClassesClassDescription像曰UDSApiDefines a class which represents the PCAN-UDS API10
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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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