《离散数学》课程教学资源（线性代数 linear algebra）英文教材PDF电子版

Linear Agebra Jim Hefferon () 1·() ) 1·1 x1·31 (⑧ ()

Linear Algebra Jim Hefferon ¡ 2 1 ¢ ¡ 1 3 ¢ ¯ ¯ ¯ ¯ 1 2 3 1 ¯ ¯ ¯ ¯ ¡ 2 1 ¢ x1 · ¡ 1 3 ¢ ¯ ¯ ¯ ¯ x1 · 1 2 x1 · 3 1 ¯ ¯ ¯ ¯ ¡ 2 1 ¢ ¡ 6 8 ¢ ¯ ¯ ¯ ¯ 6 2 8 1 ¯ ¯ ¯ ¯

Notation R.R+.Rn real numbers,reals greater than 0,n-tuples of reals W natural numbers:{0,1,2,...} C complex numbers {….} set of...such that... (a.b),[a.b interval (open or closed)of reals between a and b (.） sequence;like a set but order matters V.W.U vector spaces 可，而 vectors 0,Ov zero vector,zero vector of V B.D bases En =(e1,...,en) standard basis for Rm 6.6 basis vectors RepB(） matrix representing the vector Pn set of n-th degree polynomials Mnxm set of nxm matrices S] span of the set S M⊕N direct sum of subspaces V≌W isomorphic spaces h,g homomorphisms,linear maps H.G matrices t.s transformations;maps from a space to itself T,S square matrices RepB.D(h) matrix representing the map h hij matrix entry from row i,columnj determinant of the matrix T 冤(h),(h) rangespace and nullspace of the map h R(h),(h) generalized rangespace and nullspace Lower case Greek alphabet name character name character name character alpha iota rho p beta S kappa sigma 0 gamma 八 lambda 入 tau T delta 6 mu L upsilon epsilon nu phi 中 zeta xi E chi X eta 0 omicron 0 psi ψ theta pi omega Cover.This is Cramer's Rule for the system +2x2=6,3x1+2=8.The size of the first box is the determinant shown (the absolute value of the size is the area).The size of the second box is ri times that,and equals the size of the final box.Hence, is the final determinant divided by the first determinant

Notation R, R +, R n real numbers, reals greater than 0, n-tuples of reals N natural numbers: {0, 1, 2, . . .} C complex numbers {. . . ¯ ¯ . . .} set of . . . such that . . . (a .. b), [a .. b] interval (open or closed) of reals between a and b h. . .i sequence; like a set but order matters V, W, U vector spaces ~v, ~w vectors ~0, ~0V zero vector, zero vector of V B, D bases En = h~e1, . . . , ~eni standard basis for R n β, ~ ~δ basis vectors RepB(~v) matrix representing the vector Pn set of n-th degree polynomials Mn×m set of n×m matrices [S] span of the set S M ⊕ N direct sum of subspaces V ∼= W isomorphic spaces h, g homomorphisms, linear maps H, G matrices t, s transformations; maps from a space to itself T, S square matrices RepB,D(h) matrix representing the map h hi,j matrix entry from row i, column j |T| determinant of the matrix T R(h), N (h) rangespace and nullspace of the map h R∞(h), N∞(h) generalized rangespace and nullspace Lower case Greek alphabet name character name character name character alpha α iota ι rho ρ beta β kappa κ sigma σ gamma γ lambda λ tau τ delta δ mu µ upsilon υ epsilon ² nu ν phi φ zeta ζ xi ξ chi χ eta η omicron o psi ψ theta θ pi π omega ω Cover. This is Cramer’s Rule for the system x1 + 2x2 = 6, 3x1 + x2 = 8. The size of the first box is the determinant shown (the absolute value of the size is the area). The size of the second box is x1 times that, and equals the size of the final box. Hence, x1 is the final determinant divided by the first determinant

Preface This book helps students to master the material of a standard undergraduate linear algebra course. The material is standard in that the topics covered are Gaussian reduction, vector spaces,linear maps,determinants,and eigenvalues and eigenvectors.The audience is also standard:sophmores or juniors,usually with a background of at least one semester of Calculus and perhaps with as much as three semesters. The help that it gives to students comes from taking a developmental ap- proach-this book's presentation emphasizes motivation and naturalness,driven home by a wide variety of examples and extensive,careful,exercises.The de- velopmental approach is what sets this book apart,so some expansion of the term is appropriate here. Courses in the beginning of most Mathematics programs reward students less for understanding the theory and more for correctly applying formulas and algorithms.Later courses ask for mathematical maturity:the ability to follow different types of arguments,a familiarity with the themes that underly many mathematical investigations like elementary set and function facts,and a capac- ity for some independent reading and thinking.Linear algebra is an ideal spot to work on the transistion between the two kinds of courses.It comes early in a program so that progress made here pays off later,but also comes late enough that students are often majors and minors.The material is coherent,accessible and elegant.There are a variety of argument styles-proofs by contradiction, if and only if statements,and proofs by induction,for instance-and examples are plentiful. So.the aim of this book's exposition is to help students develop from being successful at their present level,in classes where a majority of the members are interested mainly in applications in science or engineering,to being successful at the next level,that of serious students of the subject of mathematics itself. Helping students make this transition means taking the mathematics seri- ously,so all of the results in this book are proved.On the other hand,we cannot assume that students have already arrived,and so in contrast with more abstract texts,we give many examples and they are often quite detailed. In the past,linear algebra texts commonly made this transistion abrubtly. They began with extensive computations of linear systems,matrix multiplica- tions,and determinants.When the concepts-vector spaces and linear maps- finally appeared,and definitions and proofs started,often the change brought students to a stop.In this book,while we start with a computational topic, iiⅲ

Preface This book helps students to master the material of a standard undergraduate linear algebra course. The material is standard in that the topics covered are Gaussian reduction, vector spaces, linear maps, determinants, and eigenvalues and eigenvectors. The audience is also standard: sophmores or juniors, usually with a background of at least one semester of Calculus and perhaps with as much as three semesters. The help that it gives to students comes from taking a developmental ap￾proach— this book’s presentation emphasizes motivation and naturalness, driven home by a wide variety of examples and extensive, careful, exercises. The de￾velopmental approach is what sets this book apart, so some expansion of the term is appropriate here. Courses in the beginning of most Mathematics programs reward students less for understanding the theory and more for correctly applying formulas and algorithms. Later courses ask for mathematical maturity: the ability to follow different types of arguments, a familiarity with the themes that underly many mathematical investigations like elementary set and function facts, and a capac￾ity for some independent reading and thinking. Linear algebra is an ideal spot to work on the transistion between the two kinds of courses. It comes early in a program so that progress made here pays off later, but also comes late enough that students are often majors and minors. The material is coherent, accessible, and elegant. There are a variety of argument styles— proofs by contradiction, if and only if statements, and proofs by induction, for instance— and examples are plentiful. So, the aim of this book’s exposition is to help students develop from being successful at their present level, in classes where a majority of the members are interested mainly in applications in science or engineering, to being successful at the next level, that of serious students of the subject of mathematics itself. Helping students make this transition means taking the mathematics seri￾ously, so all of the results in this book are proved. On the other hand, we cannot assume that students have already arrived, and so in contrast with more abstract texts, we give many examples and they are often quite detailed. In the past, linear algebra texts commonly made this transistion abrubtly. They began with extensive computations of linear systems, matrix multiplica￾tions, and determinants. When the concepts — vector spaces and linear maps— finally appeared, and definitions and proofs started, often the change brought students to a stop. In this book, while we start with a computational topic, iii

linear reduction,from the first we do more than compute.We do linear systems quickly but completely,including the proofs needed to justify what we are com- puting.Then,with the linear systems work as motivation and at a point where the study of linear combinations seems natural,the second chapter starts with the definition of a real vector space.This occurs by the end of the third week. Another example of our emphasis on motivation and naturalness is that the third chapter on linear maps does not begin with the definition of homomor- phism,but with that of isomorphism.That's because this definition is easily motivated by the observation that some spaces are "just like"others.After that,the next section takes the reasonable step of defining homomorphism by isolating the operation-preservation idea.This approach loses mathematical slickness,but it is a good trade because it comes in return for a large gain in sensibility to students. One aim of a developmental approach is that students should feel throughout the presentation that they can see how the ideas arise,and perhaps picture themselves doing the same type of work. The clearest example of the developmental approach taken here-and the feature that most recommends this book-is the exercises.A student progresses most while doing the exercises,so they have been selected with great care.Each problem set ranges from simple checks to resonably involved proofs.Since an instructor usually assigns about a dozen exercises after each lecture,each section ends with about twice that many,thereby providing a selection.There are even a few problems that are challenging puzzles taken from various journals, competitions,or problems collections.(These are marked with a?'and as part of the fun,the original wording has been retained as much as possible.) In total,the exercises are aimed to both build an ability at,and help students experience the pleasure of,doing mathematics. Applications,and Computers.The point of view taken here,that linear algebra is about vector spaces and linear maps,is not taken to the complete ex- clusion of others.Applications and the role of the computer are important and vital aspects of the subject.Consequently,each of this book's chapters closes with a few application or computer-related topics.Some are:network flows,the speed and accuracy of computer linear reductions,Leontief Input/Output anal- ysis,dimensional analysis,Markov chains,voting paradoxes,analytic projective geometry,and difference equations. These topics are brief enough to be done in a day's class or to be given as independent projects for individuals or small groups.Most simply give a reader a taste of the subject,discuss how linear algebra comes in,point to some further reading,and give a few exercises.In short,these topics invite readers to see for themselves that linear algebra is a tool that a professional must have. For people reading this book on their own.This book's emphasis on motivation and development make it a good choice for self-study.But,while a professional instructor can judge what pace and topics suit a class,if you are an independent student then perhaps you would find some advice helpful. Here are two timetables for a semester.The first focuses on core material. iv

linear reduction, from the first we do more than compute. We do linear systems quickly but completely, including the proofs needed to justify what we are com￾puting. Then, with the linear systems work as motivation and at a point where the study of linear combinations seems natural, the second chapter starts with the definition of a real vector space. This occurs by the end of the third week. Another example of our emphasis on motivation and naturalness is that the third chapter on linear maps does not begin with the definition of homomor￾phism, but with that of isomorphism. That’s because this definition is easily motivated by the observation that some spaces are “just like” others. After that, the next section takes the reasonable step of defining homomorphism by isolating the operation-preservation idea. This approach loses mathematical slickness, but it is a good trade because it comes in return for a large gain in sensibility to students. One aim of a developmental approach is that students should feel throughout the presentation that they can see how the ideas arise, and perhaps picture themselves doing the same type of work. The clearest example of the developmental approach taken here — and the feature that most recommends this book— is the exercises. A student progresses most while doing the exercises, so they have been selected with great care. Each problem set ranges from simple checks to resonably involved proofs. Since an instructor usually assigns about a dozen exercises after each lecture, each section ends with about twice that many, thereby providing a selection. There are even a few problems that are challenging puzzles taken from various journals, competitions, or problems collections. (These are marked with a ‘?’ and as part of the fun, the original wording has been retained as much as possible.) In total, the exercises are aimed to both build an ability at, and help students experience the pleasure of, doing mathematics. Applications, and Computers. The point of view taken here, that linear algebra is about vector spaces and linear maps, is not taken to the complete ex￾clusion of others. Applications and the role of the computer are important and vital aspects of the subject. Consequently, each of this book’s chapters closes with a few application or computer-related topics. Some are: network flows, the speed and accuracy of computer linear reductions, Leontief Input/Output anal￾ysis, dimensional analysis, Markov chains, voting paradoxes, analytic projective geometry, and difference equations. These topics are brief enough to be done in a day’s class or to be given as independent projects for individuals or small groups. Most simply give a reader a taste of the subject, discuss how linear algebra comes in, point to some further reading, and give a few exercises. In short, these topics invite readers to see for themselves that linear algebra is a tool that a professional must have. For people reading this book on their own. This book’s emphasis on motivation and development make it a good choice for self-study. But, while a professional instructor can judge what pace and topics suit a class, if you are an independent student then perhaps you would find some advice helpful. Here are two timetables for a semester. The first focuses on core material. iv

week Monday Wednesday Friday 1 One.I.1 One.I.1,2 One.I.2.3 2 One.I.3 One.II.1 One.II.2 3 One.III.1,2 One.III.2 Two.I.1 4 Two.I.2 Two.II Two.III.1 5 Two.III.1.2 Two.III.2 EXAM 6 Two.III.2.3 Two.III.3 Three.I.1 7 Three.I.2 Three.II.1 Three.II.2 8 Three.II.2 Three.Ⅱ.2 Three.III.l 9 Three.III.1 Three.III.2 Three.IV.1,2 10 Three.IV.2,3,4 Three.IV.4 EXAM 11 Three.IV.4,Three.V.1 Three.V.1,2 Four.I.1.2 Four.I.3 Four.IⅡ Four.II 13 Four.III.1 Five.I Five.II.l 14 Five.II.2 Five.II.3 REVIEW The second timetable is more ambitious(it supposes that you know One.II,the elements of vectors,usually covered in third semester calculus). week Monday Wednesday Friday 1 One.I.1 One.I.2 One.I.3 2 One.I.3 One.III.1.2 One.IⅡ.2 3 Two.I.l Two.I.2 Two.II 4 Two.III.1 Two.III.2 Two.IIL.3 5 Two.III.4 Three.I.1 EXAM 6 Three.I.2 Three.II.1 Three.II.2 7 Three.III.1 Three.III.2 Three.IV.1,2 8 Three.IV.2 Three.IV.3 Three.IV.4 9 Three.V.1 Three.V.2 Three.VI.1 10 Three.VI.2 Four.I.1 EXAM 11 Four.I.2 Four.I.3 Four.I.4 12 Four.II Four.II,Four.III.1 Four.III.2.3 13 Five.II.1,2 Five.II.3 Five.III.1 14 Five.III.2 Five.IV.1.2 Five.IV.2 See the table of contents for the titles of these subsections. To help you make time trade-offs,in the table of contents I have marked sub- sections as optional if some instructors will pass over them in favor of spending more time elsewhere.You might also try picking one or two topics that appeal to you from the end of each chapter.You'll get more from these if you have access to computer software that can do the big calculations. The most important advice is:do many exercises.I have marked a good sample with v's.(The answers are available.)You should be aware,however, that few inexperienced people can write correct proofs.Try to find a knowl- edgeable person to work with you on this. Finally,if I may,a caution for all students,independent or not:I cannot overemphasize how much the statement that I sometimes hear,"I understand

week Monday Wednesday Friday 1 One.I.1 One.I.1, 2 One.I.2, 3 2 One.I.3 One.II.1 One.II.2 3 One.III.1, 2 One.III.2 Two.I.1 4 Two.I.2 Two.II Two.III.1 5 Two.III.1, 2 Two.III.2 exam 6 Two.III.2, 3 Two.III.3 Three.I.1 7 Three.I.2 Three.II.1 Three.II.2 8 Three.II.2 Three.II.2 Three.III.1 9 Three.III.1 Three.III.2 Three.IV.1, 2 10 Three.IV.2, 3, 4 Three.IV.4 exam 11 Three.IV.4, Three.V.1 Three.V.1, 2 Four.I.1, 2 12 Four.I.3 Four.II Four.II 13 Four.III.1 Five.I Five.II.1 14 Five.II.2 Five.II.3 review The second timetable is more ambitious (it supposes that you know One.II, the elements of vectors, usually covered in third semester calculus). week Monday Wednesday Friday 1 One.I.1 One.I.2 One.I.3 2 One.I.3 One.III.1, 2 One.III.2 3 Two.I.1 Two.I.2 Two.II 4 Two.III.1 Two.III.2 Two.III.3 5 Two.III.4 Three.I.1 exam 6 Three.I.2 Three.II.1 Three.II.2 7 Three.III.1 Three.III.2 Three.IV.1, 2 8 Three.IV.2 Three.IV.3 Three.IV.4 9 Three.V.1 Three.V.2 Three.VI.1 10 Three.VI.2 Four.I.1 exam 11 Four.I.2 Four.I.3 Four.I.4 12 Four.II Four.II, Four.III.1 Four.III.2, 3 13 Five.II.1, 2 Five.II.3 Five.III.1 14 Five.III.2 Five.IV.1, 2 Five.IV.2 See the table of contents for the titles of these subsections. To help you make time trade-offs, in the table of contents I have marked sub￾sections as optional if some instructors will pass over them in favor of spending more time elsewhere. You might also try picking one or two topics that appeal to you from the end of each chapter. You’ll get more from these if you have access to computer software that can do the big calculations. The most important advice is: do many exercises. I have marked a good sample with X’s. (The answers are available.) You should be aware, however, that few inexperienced people can write correct proofs. Try to find a knowl￾edgeable person to work with you on this. Finally, if I may, a caution for all students, independent or not: I cannot overemphasize how much the statement that I sometimes hear, “I understand v