实例介绍
【实例简介】
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。
PREFACE My purpose in this book is to treat linear transformations on finite- dimensional vector spaces by the methods of more general theories. The idea is to emphasize the simple geometric notions common to many parts of mathematics and its applications, and to do so in a language that gives away the trade secrets and tells the student what is in the back of the minds of people proving theorems about integral equations and Hilbert spaces. The reader does not, however, have to share my prejudiced motivation Except for an occasional reference to undergraduate mathematics the book is self-contained and may be read by anyone who is trying to get a feeling for the linear problems usually discussed in courses on matrix theory or higher"algebra. The algebraic, coordinate-free methods do not lose power and elegance by specialization to a finite number of dimensions, and they are, in my belief, as elementary as the classical coordinatized treatment I originally intended this book to contain a theorem if and only if an infinite-dimensional generalization of it already exists, The tempting easiness of some essentially finite-dimensional notions and results was however, irresistible, and in the final result my initial intentions are just barely visible. They are most clearly seen in the emphasis, throughout, on generalizable methods instead of sharpest possible results. The reader may sometimes see some obvious way of shortening the proofs i give In such cases the chances are that the infinite-dimensional analogue of the shorter proof 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)has been circulating for several years. In addition to some minor changes in style and in order, the difference between the preceding version and this one 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 products special attention to the real case.(2)a definition of determinants in invariant terms, via the theory of multilinear forms. 3 Exercises The exercises(well over three hundred of them) constitute the most significant addition; I hope that they will be found useful by both student PREFACE and teacher. There are two things about them the reader should know First, if an exercise is neither imperative "prove that.., )nor interrog tive("is it true that...?" )but merely declarative, then it is intended as a challenge. For such exercises the reader is asked to discover if the assertion is true or false, prove it if true and construct a counterexample if false, and, most important of all, discuss such alterations of hypothesis and conclusion as will make the true ones false and the false ones true. Second the exercises, whatever their grammatical form, are not always placed 8o as 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 before machinery 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 for his attempt. Having in mind possible future editions of the book, I ask the 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 been known 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 papers from which I learned and to the friends and strangers who, before and after the publication of the first version, gave me much valuable encourage- ment and criticism. I am particularly grateful to three men: J. L. Doob and arlen Brown, who read the entire manuscript of the first and the second version, respectively, and made many useful suggestions, and John von Neumann, who was one of the originators of the modern spirit and methods that I have tried to present and whose teaching was the inspiration for this book P、R.H CONTENTS 的 FAPTER PAGR I SPACES I. Fields, 1; 2. Vector spaces, 3; 3. Examples, 4;4. Comments, 5 5. Linear dependence, 7; 6. Linear combinations. 9: 7. Bases, 10 8. Dimension, 13; 9. Isomorphism, 14; 10. Subspaces, 16; 11. Cal culus of subspaces, 17; 12. Dimension of a subspace, 18; 13. Dual spaces, 20; 14. Brackets, 21; 15. Dual bases, 23; 16. Reflexivity, 24; 17. Annihilators, 26; 18. Direct sums, 28: 19. Dimension of a direct sum, 30; 20. Dual of a direct sum, 31; 21. Q guotient spaces, 33; 22. Dimension of a quotient space, 34; 23. Bilinear forms, 35 24. Tensor products, 38; 25. Product bases, 40 26. Permutations 41; 27. Cycles, 44; 28. Parity, 46; 29. Multilinear forms, 48 30. Alternating formB, 50; 31. Alternating forms of maximal degree, 52 II. TRANSFORMATIONS 32. Linear transformations, 55; 33. Transformations as vectors, 56 34. 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. Similar ity, 84; 48. Quotient transformations, 87; 49. Range and null- space, 88; 50. Rank and nullity, 90; 51. Transformations of rank one, 92 52. Tensor products of transformations, 95; 53. Determi nants, 98 54. Proper values, 102; 55. Multiplicity, 104; 56. Tri angular form, 106; 57. Nilpotence, 109; 58. Jordan form. 112 III ORTHOGONALITY 118 59. Inner products, 118; 60. Complex inner products, 120; 61. Inner product spaces, 121; 62 Orthogonality, 122; 63. Completeness, 124; 64. Schwarz e inequality, 125; 65. Complete orthonormal sets, 127; CONTENTS 66. Projection theorem, 129; 67. Linear functionals, 130; 68. P aren, gB CHAPTER theses versus brackets, 131 69. Natural isomorphisms, 138; 70. Self-adjoint transformations, 135: 71. Polarization, 138 72. Positive transformations, 139; 73. Isometries, 142; 74. Change of orthonormal basis, 144; 75. Perpendicular projections, 146 76. Combinations of perpendicular projections, 148; 77. Com- plexification, 150; 78. Characterization of spectra, 158; 79. Spec- p tral theorem, 155; 80. normal transformations, 159; 81. Orthogonal transformations, 162; 82. Functions of transformations, 165 83. Polar decomposition, 169; 84. Commutativity, 171; 85. Self- adjoint transformations of rank one, 172 IV. ANALYSIS.... 175 86. Convergence of vectors, 175; 87. Norm, 176; 88. Expressions for the norm, 178; 89. bounds of a self-adjoint transformation, 179 90. Minimax principle, 181; 91. Convergence of linear transforma tions, 182 92. Ergodic theorem, 184 98. Power series, 186 APPENDIX. HILBERT SPACE RECOMMENDED READING, 195 INDEX OF TERMS, 197 INDEX OF SYMBOLS, 200 CHAPTER I SPACES §L. Fields In 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 specific class, we shall adopt the dodge of referring to numbers as scalars. The reader will not lose anything essential if he consistently interprets scalars as real numbers or as complex numbers in the examples that we shall study both classes will occur. To be specific(and also in order to operate at the proper level of generality) we proceed to list all the general facts about 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 for every 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 aB called 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 a for every scalar a, and (4)to every non-zero scalar a there corresponds a unique scalar a-1 or-such that aa SPACES (C)Multiplication is distributive with respect to addition, a(a+n) If addition and multiplication are defined within some set of objects scalars) so that the conditions(A),B), and (c)are satisfied, then that set(together with the given operations) is called a field. Thus, for example the set Q of all rational numbers(with the ordinary definitions of sum and product)is a field, and the same is true of the set of all real numbera and the set e of all complex numbers HHXERCISIS 1. Almost all the laws of elementary arithmetic are consequences of the axioms defining a field. Prove, in particular, that if 5 is field and if a, and y belong to 5. then the following relations hold 80+a=a b )Ifa+B=a+r, then p=y ca+(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 the integers, we shall almost always use the ordinary operations of addition and multi- lication. On the rare occasions when we depart from this convention, we shall give ample warning As for "positive, "by that word we mean, here and elsewhere in 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 or multiplication (or both)? 3. Let m be an integer, m2 2, and let Zm be the set of all positive integers less than m, zm=10, 1, .. m-1). If a and B are in Zmy let a +p be the least positive remainder obtained by dividing the(ordinary) sum of a and B by m, and product 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 of elementary arithmetic hold in fields; in Z2, for instance, 1 +1 =0. Prove that if 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 number of summands is a prime. (The characteristic of the field s is defined to be 0 in the first case and the crucial prime in the second) SEC. 2 VECTOR SPACES 3 5. Let Q(v2)be the set of all real numbers of the form a+Bv2, where a 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) d aB)(,b)=(ay-6a6+的y), is g a field then? (c)What happens (in both the preceding cases)if we consider ordered pairs of complex numbers instead? §2. Vector space We come now to the basic concept of this book. For the definition that follows we assume that we are given a particular field s; the scalars to be used are to be elements of g DEFINITION. A vector space is a set o of elements called vectors satisfying the following axioms Q (A)To every pair, a and g, of vectors in u there corresponds vector a 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 that a t0=s for every vector and (4)to every vector r in U there corresponds a unique vector -r that 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 such a way that (1)multiplication by scalars is associative, a(Bx)=aB)=, and (2 lz a s for every vector x SPACES SFC B (C)(1)Multiplication by scalars is distributive with respect to vector ddition, a(+y=a+ ag, and 2)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 are merely a convenient characterization of the objects we wish to study. The relation between a vector space V and the underlying field s is usually described by saying that v is a vector space over 5. If S is the field R of real number, u is called a real vector space; similarly if s is Q or if g ise, we speak of rational vector spaces or complex vector space §3. Examples Before discussing the implications of the axioms, we give some examples We shall refer to these examples over and over again, and we shall use the notation established here throughout the rest of our work. (1) Let e(= e)be the set of all complex numbers; if we interpret r+y and az as ordinary complex numerical addition and multiplication e becomes a complex vector space 2)Let o be the set of all polynomials, with complex coeficients, in a variable t. To make into a complex vector space, we interpret vector addition and scalar multiplication as the ordinary addition of two poly- nomials and the multiplication of a polynomial by a complex number the origin in o is the polynomial identically zero Example(1)is too simple and example (2)is too complicated to be typical of the main contents of this book. We give now another example of complex vector spaces which(as we shall see later)is general enough for all 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,,b definition z+y=〔1+叽,…十物m) 0=(0,…,0), -in It is easy to verify that all parts of our axioms(a),(B), and (C),52, are satisfied, so that en is a complex vector space; it will be called n-dimenaional complex coordinate space 【实例截图】
【核心代码】
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。
PREFACE My purpose in this book is to treat linear transformations on finite- dimensional vector spaces by the methods of more general theories. The idea is to emphasize the simple geometric notions common to many parts of mathematics and its applications, and to do so in a language that gives away the trade secrets and tells the student what is in the back of the minds of people proving theorems about integral equations and Hilbert spaces. The reader does not, however, have to share my prejudiced motivation Except for an occasional reference to undergraduate mathematics the book is self-contained and may be read by anyone who is trying to get a feeling for the linear problems usually discussed in courses on matrix theory or higher"algebra. The algebraic, coordinate-free methods do not lose power and elegance by specialization to a finite number of dimensions, and they are, in my belief, as elementary as the classical coordinatized treatment I originally intended this book to contain a theorem if and only if an infinite-dimensional generalization of it already exists, The tempting easiness of some essentially finite-dimensional notions and results was however, irresistible, and in the final result my initial intentions are just barely visible. They are most clearly seen in the emphasis, throughout, on generalizable methods instead of sharpest possible results. The reader may sometimes see some obvious way of shortening the proofs i give In such cases the chances are that the infinite-dimensional analogue of the shorter proof 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)has been circulating for several years. In addition to some minor changes in style and in order, the difference between the preceding version and this one 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 products special attention to the real case.(2)a definition of determinants in invariant terms, via the theory of multilinear forms. 3 Exercises The exercises(well over three hundred of them) constitute the most significant addition; I hope that they will be found useful by both student PREFACE and teacher. There are two things about them the reader should know First, if an exercise is neither imperative "prove that.., )nor interrog tive("is it true that...?" )but merely declarative, then it is intended as a challenge. For such exercises the reader is asked to discover if the assertion is true or false, prove it if true and construct a counterexample if false, and, most important of all, discuss such alterations of hypothesis and conclusion as will make the true ones false and the false ones true. Second the exercises, whatever their grammatical form, are not always placed 8o as 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 before machinery 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 for his attempt. Having in mind possible future editions of the book, I ask the 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 been known 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 papers from which I learned and to the friends and strangers who, before and after the publication of the first version, gave me much valuable encourage- ment and criticism. I am particularly grateful to three men: J. L. Doob and arlen Brown, who read the entire manuscript of the first and the second version, respectively, and made many useful suggestions, and John von Neumann, who was one of the originators of the modern spirit and methods that I have tried to present and whose teaching was the inspiration for this book P、R.H CONTENTS 的 FAPTER PAGR I SPACES I. Fields, 1; 2. Vector spaces, 3; 3. Examples, 4;4. Comments, 5 5. Linear dependence, 7; 6. Linear combinations. 9: 7. Bases, 10 8. Dimension, 13; 9. Isomorphism, 14; 10. Subspaces, 16; 11. Cal culus of subspaces, 17; 12. Dimension of a subspace, 18; 13. Dual spaces, 20; 14. Brackets, 21; 15. Dual bases, 23; 16. Reflexivity, 24; 17. Annihilators, 26; 18. Direct sums, 28: 19. Dimension of a direct sum, 30; 20. Dual of a direct sum, 31; 21. Q guotient spaces, 33; 22. Dimension of a quotient space, 34; 23. Bilinear forms, 35 24. Tensor products, 38; 25. Product bases, 40 26. Permutations 41; 27. Cycles, 44; 28. Parity, 46; 29. Multilinear forms, 48 30. Alternating formB, 50; 31. Alternating forms of maximal degree, 52 II. TRANSFORMATIONS 32. Linear transformations, 55; 33. Transformations as vectors, 56 34. 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. Similar ity, 84; 48. Quotient transformations, 87; 49. Range and null- space, 88; 50. Rank and nullity, 90; 51. Transformations of rank one, 92 52. Tensor products of transformations, 95; 53. Determi nants, 98 54. Proper values, 102; 55. Multiplicity, 104; 56. Tri angular form, 106; 57. Nilpotence, 109; 58. Jordan form. 112 III ORTHOGONALITY 118 59. Inner products, 118; 60. Complex inner products, 120; 61. Inner product spaces, 121; 62 Orthogonality, 122; 63. Completeness, 124; 64. Schwarz e inequality, 125; 65. Complete orthonormal sets, 127; CONTENTS 66. Projection theorem, 129; 67. Linear functionals, 130; 68. P aren, gB CHAPTER theses versus brackets, 131 69. Natural isomorphisms, 138; 70. Self-adjoint transformations, 135: 71. Polarization, 138 72. Positive transformations, 139; 73. Isometries, 142; 74. Change of orthonormal basis, 144; 75. Perpendicular projections, 146 76. Combinations of perpendicular projections, 148; 77. Com- plexification, 150; 78. Characterization of spectra, 158; 79. Spec- p tral theorem, 155; 80. normal transformations, 159; 81. Orthogonal transformations, 162; 82. Functions of transformations, 165 83. Polar decomposition, 169; 84. Commutativity, 171; 85. Self- adjoint transformations of rank one, 172 IV. ANALYSIS.... 175 86. Convergence of vectors, 175; 87. Norm, 176; 88. Expressions for the norm, 178; 89. bounds of a self-adjoint transformation, 179 90. Minimax principle, 181; 91. Convergence of linear transforma tions, 182 92. Ergodic theorem, 184 98. Power series, 186 APPENDIX. HILBERT SPACE RECOMMENDED READING, 195 INDEX OF TERMS, 197 INDEX OF SYMBOLS, 200 CHAPTER I SPACES §L. Fields In 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 specific class, we shall adopt the dodge of referring to numbers as scalars. The reader will not lose anything essential if he consistently interprets scalars as real numbers or as complex numbers in the examples that we shall study both classes will occur. To be specific(and also in order to operate at the proper level of generality) we proceed to list all the general facts about 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 for every 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 aB called 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 a for every scalar a, and (4)to every non-zero scalar a there corresponds a unique scalar a-1 or-such that aa SPACES (C)Multiplication is distributive with respect to addition, a(a+n) If addition and multiplication are defined within some set of objects scalars) so that the conditions(A),B), and (c)are satisfied, then that set(together with the given operations) is called a field. Thus, for example the set Q of all rational numbers(with the ordinary definitions of sum and product)is a field, and the same is true of the set of all real numbera and the set e of all complex numbers HHXERCISIS 1. Almost all the laws of elementary arithmetic are consequences of the axioms defining a field. Prove, in particular, that if 5 is field and if a, and y belong to 5. then the following relations hold 80+a=a b )Ifa+B=a+r, then p=y ca+(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 the integers, we shall almost always use the ordinary operations of addition and multi- lication. On the rare occasions when we depart from this convention, we shall give ample warning As for "positive, "by that word we mean, here and elsewhere in 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 or multiplication (or both)? 3. Let m be an integer, m2 2, and let Zm be the set of all positive integers less than m, zm=10, 1, .. m-1). If a and B are in Zmy let a +p be the least positive remainder obtained by dividing the(ordinary) sum of a and B by m, and product 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 of elementary arithmetic hold in fields; in Z2, for instance, 1 +1 =0. Prove that if 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 number of summands is a prime. (The characteristic of the field s is defined to be 0 in the first case and the crucial prime in the second) SEC. 2 VECTOR SPACES 3 5. Let Q(v2)be the set of all real numbers of the form a+Bv2, where a 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) d aB)(,b)=(ay-6a6+的y), is g a field then? (c)What happens (in both the preceding cases)if we consider ordered pairs of complex numbers instead? §2. Vector space We come now to the basic concept of this book. For the definition that follows we assume that we are given a particular field s; the scalars to be used are to be elements of g DEFINITION. A vector space is a set o of elements called vectors satisfying the following axioms Q (A)To every pair, a and g, of vectors in u there corresponds vector a 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 that a t0=s for every vector and (4)to every vector r in U there corresponds a unique vector -r that 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 such a way that (1)multiplication by scalars is associative, a(Bx)=aB)=, and (2 lz a s for every vector x SPACES SFC B (C)(1)Multiplication by scalars is distributive with respect to vector ddition, a(+y=a+ ag, and 2)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 are merely a convenient characterization of the objects we wish to study. The relation between a vector space V and the underlying field s is usually described by saying that v is a vector space over 5. If S is the field R of real number, u is called a real vector space; similarly if s is Q or if g ise, we speak of rational vector spaces or complex vector space §3. Examples Before discussing the implications of the axioms, we give some examples We shall refer to these examples over and over again, and we shall use the notation established here throughout the rest of our work. (1) Let e(= e)be the set of all complex numbers; if we interpret r+y and az as ordinary complex numerical addition and multiplication e becomes a complex vector space 2)Let o be the set of all polynomials, with complex coeficients, in a variable t. To make into a complex vector space, we interpret vector addition and scalar multiplication as the ordinary addition of two poly- nomials and the multiplication of a polynomial by a complex number the origin in o is the polynomial identically zero Example(1)is too simple and example (2)is too complicated to be typical of the main contents of this book. We give now another example of complex vector spaces which(as we shall see later)is general enough for all 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,,b definition z+y=〔1+叽,…十物m) 0=(0,…,0), -in It is easy to verify that all parts of our axioms(a),(B), and (C),52, are satisfied, so that en is a complex vector space; it will be called n-dimenaional complex coordinate space 【实例截图】
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