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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2017年深圳杯建模挑战赛C题决赛论文（独家）
本人2017年入选了深圳杯建模决赛，这是正式答辩时的论文。仅供各位建模的同学学习参考。基于问趣二中的预测结果,计算得出未来十年的总成本数量及各模式下各分项成本(涵盖直接业务成本、经济技术成本、问接的当下和远期社会成本)比例,分析其变化趋势,并通过作图直观反映。考虑到目前深圳已经开始建立生活垃吸强制分类制度,本文详细分析了家庭分类与专业分类两种前端分流模式对总成本的影响。问题三的分析基于较为完善的模型,对远期效益成木比进行估算,从深圳市具休情况岀发,设计一套生活垃圾处理的优选模式,以供参考。符号说明变量含义生活垃圾处理社会总成本直接成本直接业务成本收集成本第种收集方式的成本第种收集方式中的第种成本运输成木第种运输方式的成本第种运输方式中的第种成本处理成本第种处理方式成本绎济技术成本固定成本第种固定成本可变成本第种可变成本税收减免第种税收减免间接的当下和远期社会成本,即环境损失成本第种环境损失成本年度垃圾处理量湿垃圾占垃圾总量的比例垃圾分类中的干垃圾总量垃圾分类中的湿垃圾总量模型假设假设在估算时间内国家及深圳市相关政策不变。假设在估算时间内折现※为假设在佔算时间内政府对源头分类的补贴保持不变倀设在估算时间内填埋、焚烧、生物处理三种方式下的基准地价的季度增长率分别为模型建立与求解问题一的模型建立模型的准备针对问题一,将生活垃圾处理的社会总成本分为直接业务成本、经济技术成本、问接的当下和远期社会成本,如图生活垃圾处理社会总成本直接业务成本D经济技术成本E间接的当下和远期社会成本图生活垃圾处理社会总成本构成直接业务成本分析在直接业务成木中,我们又将其细化分成了垃圾收集成木、运输成木和处理成本三个部分,如图直接业务成本D收集成本D处理成本D匚运输成本D图直接业务成本构成4()收集成本分析由附件一可知,在不同的垃圾处理模式中,收集方式分别对应混合收集、源头分类收集和混合收集末端分类。即收集成本可划分为混合收集成本源头分类收集成本及混合收集木端分类成本而且每一种收集方式的成本又涵盖了公用垃圾桶成本(分别对应)和运输成本(分别对应),值得注意的是,源头分类收集方式会有额外的政府补贴,而混合收集末端分类方式在末端分类时会占用额外的土地、人力、设备等,因此会产生额外的成本和,如图收集成本De混合收集源头分类成本收集成本集,末端分类成本D公用福政府Dcg成本图收集成本构成()运输成本分析运输成本分为混合运输成本和分类运输成本两类,其中每一种运输成本都包括转运站成本(表示从各公用桶运输到转运站进行进一步处理所需成本,分别对应)和运输成本(分别对应),如图:运输成本Dt混合分类运输运输成本成本转运站成运输站成运输本成本DtD图运输成本构成5()处理成本分析由于处理模式的不同,处理成本可分为焚烧处理成本、填埋处理成本以及生物处理成本,如图处理成本D。焚烧填埋生物处理处理处理成本成本成本DstD图处理成本构成经济技术成本分析经济技术成本包括固定成本、可变成本和税收减免。固定成本分为土地成本和建设成本可变成本包括飞灰补贴、底灰补贴电价补贴、渗沥液补贴以及其他补贴;税收减笕分为増值税减笕、营业税减免和企业所得税减免,如图:经济技术成本E税收减免可变成本定成本E稅减税减税减其他补赎二c5EC2填埋场生物处理厂图经济技术成本构成间接的当下和远期社会成本分析问接的当下和远期成本涵盖了由于环保标准提高所花费的成本、水污染造成的损失、大气污染造成的损失以及固体废弁物污染造成的损失如图6问接的当下和远期社会成本L水污染损失气污染固体废物污染损失L4业损幻1匚人体健康损失[林损头「其他损类图间接的当下和远期社会成本构成模型的建立考虑到不同情况下,决策者会选择不同的模式组合方式。因此,在计算社会总成本时,本文选用各个模式下不同环节所需成本的叠加,从而求得深圳市生活垃圾处理的社会总成木。直接业务成本的计算直接业务成本包括收集成本、运输成本、处理成本直接业务成本的计算公式为:()收集成本的计算公式为=∑∑++代表混合收集成本,代表源头分类收集成本,代表源头混合收集,末端分类成木;代表第种收集方式的公用桶成木,代表第种收集方式的运输成本,代表单位垃圾消耗的公用桶成本,代表生活垃圾年产量,此成本仅包括垃圾桶至小型转运站的成本,如果决策者选择源头混合收集,末端分类方式收集垃圾,则应该加上额外的土地、设各、人工等成本(后面统称为额外成本),如果选择源头分类,则应加上政府补贴;代表单位吨数单位公里运输价格(是一个与距离有关的分段函数),代表距离;代表单位湿垃圾政府补贴成木;代表单位土地、设备、人工等的成木代表湿垃圾占总垃圾量的比重。()运输成本包括混合运输成本、分类运输成本运输成本的计算公式为7代表第种运输方式的转运站成本,包括转运站人工费,以及设备维护费等,代表第科运输方式的运输成本,此成本仅包括小型转运站至末端垃圾处理站的成本,代表单位转运站成本。()处理成本包括焚烧成本、填埋成本和生物处理成本处理成木的计算公式为代表单位垃圾焚烧处理成本;代表单位垃圾填埋处理成木;代表单位垃圾生物处理成本。经济技术成本的计算经济技术成本包括固定成本、可变成本、税收减免经济技术成木的计算公式为固定成本包括焚烧垃圾的十地成本、填埋的十地成本、建设成木。固定成木的计算公式为:代表当年地价,代表十地面积,代表折现率,代表工业用地年;十地机会成本为;代表年地价,代表季度地价增长率,代表时间:代表填哩高度;ρ代表填埋密度代表十地机会成本;代表建设补贴。可变成本包括飞灰补贴、底灰补贴、电价补贴渗沥液补可变成本的计算公式为8=代表单位底灰处理补贴,代表底灰量,代表单位飞灰处理补贴,代表K灰量;代表上网电价补贴,代表超额供电补贴;代表单位污水处理补贴,代表污水处理量税收减免包括增值税减免、营业税减免企业所得税减免税收减免的计算公式为:间接的当下和远期社会成本的计算间接的当下和远期社会成本包括环保标准提高后所需成本(远期环境标准提髙后垃圾处理费升高所需成本)、水污染导致的健康损失、空气污染健康损失、固体废弃物污染损失。水污染导致的健康损失包括早逝引起的健康损失、疾病治疗费用和误工损失;由于垃圾厂排放的气休中对人体造成巨大损失的气体为二嵁英,故将空气污染健康损失考虑为二嗯英造成的健康损失;在计算固休废弃物污染时,采用市场价值法对生活垃圾固休废弃物造成的人工管理费、设备费和运输费等费用进行计算不管对生活垃圾使用哪种处理方式,在计算生活垃圾堆存造成的经济损失时,以需按填埋量来进行计算间接的当下和远期社会成本为:∑∑总数总数指标评价和危险特征阐述,结合国际组织的研究成果,对水污染健康损伤的不良影响进行定量评{,代表年人均收入,代表就诊人数,代表人均治疗费用,代表日均收入,代表住院病例,代表每患病者入住天数;代表不同浓度区域的编码,代表不同浓度区域的二惡英致癌风险代表每平方公里人口密度,代表不同浓度区域所占的面积,代表个体生命价值代表治疗费用;代表生活垃圾堆存损失系数,代表生活垃圾堆存量。综上,由上述各式可得生活垃圾的社会总成本为:+十问题二模型建立及解决各模式的直接成本估算方案完善深圳市生活垃圾直接成木包括直接业务成木和经济技术成木,由式()至式()可知,城市生活垃圾直接成本为:当期社会总成本估算由已知数据代入问题一模型中可知年的社会总成本为:+十元日前仅得到年深圳市生活垃圾年产量数据,如图:s0047544069406图年深圳市生活垃圾年产量无法直接计算出当期以及未来十年各模式下直接成本,故基于灰色预测方法,根捃此数据,估算得出年的生活垃圾年产量如表
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