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 approach— this book’s presentation emphasizes motivation and naturalness, driven home by a wide variety of examples and extensive, careful, exercises. The developmental 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 capacity 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 seriously, 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 multiplications, 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 computing. 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 homomorphism, 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 exclusion 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 analysis, 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 subsections 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 knowledgeable 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
the material,but it's only that I have trouble with the problems"reveals a lack of understanding of what we are up to.Being able to do things with the ideas is their point.The quotes below express this sentiment admirably.They state what I believe is the key to both the beauty and the power of mathematics and the sciences in general,and of linear algebra in particular(I took the liberty of formatting them as poems). I know of no better tactic than the illustration of exciting principles by well-chosen particulars. -Stephen Jay Gould If you really wish to learn then you must mount the machine and become acquainted with its tricks by actual trial. -Wilbur Wright Jim Hefferon Mathematics,Saint Michael's College Colchester,Vermont USA 05439 http://joshua.smcvt.edu 2006-May-20 Author's Note.Inventing a good exercise,one that enlightens as well as tests, is a creative act,and hard work.The inventor deserves recognition.But for some reason texts have traditionally not given attributions for questions.I have changed that here where I was sure of the source.I would greatly appreci- ate hearing from anyone who can help me to correctly attribute others of the questions. vi
the material, but it’s only that I have trouble with the problems” reveals a lack of understanding of what we are up to. Being able to do things with the ideas is their point. The quotes below express this sentiment admirably. They state what I believe is the key to both the beauty and the power of mathematics and the sciences in general, and of linear algebra in particular (I took the liberty of formatting them as poems). I know of no better tactic than the illustration of exciting principles by well-chosen particulars. –Stephen Jay Gould If you really wish to learn then you must mount the machine and become acquainted with its tricks by actual trial. –Wilbur Wright Jim Hefferon Mathematics, Saint Michael’s College Colchester, Vermont USA 05439 http://joshua.smcvt.edu 2006-May-20 Author’s Note. Inventing a good exercise, one that enlightens as well as tests, is a creative act, and hard work. The inventor deserves recognition. But for some reason texts have traditionally not given attributions for questions. I have changed that here where I was sure of the source. I would greatly appreciate hearing from anyone who can help me to correctly attribute others of the questions. vi
Contents Chapter One: Linear Systems 1 I Solving Linear Systems.················· 1 1 Gauss'Method..·················· 2 2 Describing the Solution Set....···..······ 3 General=Particular+Homogeneous。.··.······ 20 II Linear Geometry of n-Space···················· 32 1 Vectors in Space························· 32 2 Length and Angle Measures* 38 III Reduced Echelon Form.····:·················· 46 1 Gauss-Jordan Reduction.. 46 2 Row Equivalence.··.······················ Topic:Computer Algebra Systems 62 Topic:nput-Output Analysis.··· Topic:Accuracy of Computations 44 68 Topic:Analyzing Networks..... 72 Chapter Two:Vector Spaces 79 I Definition of Vector Space::···················· 80 1 Definition and Examples..,.·················· 80 2 Subspaces and Spanning Sets................... 91 IⅡLinear Independence·.. ...............101 1 Definition and Examples.·· 。 101 IⅡBasis and Dimension··..·· 112 1 Basis..··· 112 2 Dimension.。···。········· 。。 118 3 Vector Spaces and Linear Systems 123 4 Combining Subspaces*。··,··················· 130 Topic:Fields...·。..。.·。··。··。···。·。··· 140 Topic:Crystals::·::。···:·::。::·l42 Topic:Dimensional Analysis.················· ...146 vii
Contents Chapter One: Linear Systems 1 I Solving Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Gauss’ Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Describing the Solution Set . . . . . . . . . . . . . . . . . . . . 11 3 General = Particular + Homogeneous . . . . . . . . . . . . . . 20 II Linear Geometry of n-Space . . . . . . . . . . . . . . . . . . . . . 32 1 Vectors in Space . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2 Length and Angle Measures∗ . . . . . . . . . . . . . . . . . . . 38 III Reduced Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . 46 1 Gauss-Jordan Reduction . . . . . . . . . . . . . . . . . . . . . . 46 2 Row Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Topic: Computer Algebra Systems . . . . . . . . . . . . . . . . . . . 62 Topic: Input-Output Analysis . . . . . . . . . . . . . . . . . . . . . . 64 Topic: Accuracy of Computations . . . . . . . . . . . . . . . . . . . . 68 Topic: Analyzing Networks . . . . . . . . . . . . . . . . . . . . . . . . 72 Chapter Two: Vector Spaces 79 I Definition of Vector Space . . . . . . . . . . . . . . . . . . . . . . 80 1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . 80 2 Subspaces and Spanning Sets . . . . . . . . . . . . . . . . . . . 91 II Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . 101 1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . 101 III Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 112 1 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3 Vector Spaces and Linear Systems . . . . . . . . . . . . . . . . 123 4 Combining Subspaces∗ . . . . . . . . . . . . . . . . . . . . . . . 130 Topic: Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Topic: Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Topic: Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . 146 vii
Chapter Three:Maps Between Spaces 153 I Isomorphisms..,.,:·.,...。.·。··.:..·:·.,153 1 Definition and Examples.···:················· 153 2 Dimension Characterizes Isomorphism 162 II Homomorphisms 170 2 Rangespace and Nullspace.·.·················· 177 IⅡComputing Linear Maps···· 189 1 Representing Linear Maps with Matrices... 189 2 Any Matrix Represents a Linear Map*········· 199 IV Matrix Operations...·.。。·..··········· 206 1 Sums and Scalar Products.··············· 206 2 Matrix Multiplication 4 208 3 Mechanics of Matrix Multiplication············· 216 4 Inverses.·.··········· 225 V Change of Basis。·..······················ 232 1 Changing Representations of Vectors.·············· 232 2 Changing Map Representations.:....。··.。.··.··· 236 VI Projection...........···. 244 1 Orthogonal Projection Into a Line 244 2 Gram-Schmidt Orthogonalization* ..·..248 3 Projection Into a Subspace*.··················· 254 Topic:Line of Best Fit.. 263 Topic:Geometry of Linear Maps 268 Topic:Markov Chains 。。。”··” 4。 275 Topic:Orthonormal Matrices 281 Chapter Four:Determinants 287 I Definition·········,··· 288 1 Exploration* 288 2 Properties of Determinants 4 293 3 The Permutation Expansion. 44 297 4 Determinants Exist*.,.·· 306 II Geometry of Determinants.... 。。·。。。。。。。。 313 1 Determinants as Size Functions············· 313 IⅡOther Formulas.........········ 320 1 Laplace's Expansion.·,,.。·.················ 320 Topic:Cramer's Rule..·····. 325 Topic:Speed of Calculating Determinants............... 328 Topic:Projective Geometry 331 Chapter Five:Similarity 343 I Complex Vector Spaces.···.:·.················ 343 1 Factoring and Complex Numbers:A Review* 344 2 Complex Representations 345 I Similarity....··.·....·..347 viii
Chapter Three: Maps Between Spaces 153 I Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . 153 2 Dimension Characterizes Isomorphism . . . . . . . . . . . . . . 162 II Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 2 Rangespace and Nullspace . . . . . . . . . . . . . . . . . . . . . 177 III Computing Linear Maps . . . . . . . . . . . . . . . . . . . . . . . 189 1 Representing Linear Maps with Matrices . . . . . . . . . . . . . 189 2 Any Matrix Represents a Linear Map∗ . . . . . . . . . . . . . . 199 IV Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 206 1 Sums and Scalar Products . . . . . . . . . . . . . . . . . . . . . 206 2 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . 208 3 Mechanics of Matrix Multiplication . . . . . . . . . . . . . . . . 216 4 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 V Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 1 Changing Representations of Vectors . . . . . . . . . . . . . . . 232 2 Changing Map Representations . . . . . . . . . . . . . . . . . . 236 VI Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 1 Orthogonal Projection Into a Line∗ . . . . . . . . . . . . . . . . 244 2 Gram-Schmidt Orthogonalization∗ . . . . . . . . . . . . . . . . 248 3 Projection Into a Subspace∗ . . . . . . . . . . . . . . . . . . . . 254 Topic: Line of Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Topic: Geometry of Linear Maps . . . . . . . . . . . . . . . . . . . . 268 Topic: Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Topic: Orthonormal Matrices . . . . . . . . . . . . . . . . . . . . . . 281 Chapter Four: Determinants 287 I Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 1 Exploration∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 2 Properties of Determinants . . . . . . . . . . . . . . . . . . . . 293 3 The Permutation Expansion . . . . . . . . . . . . . . . . . . . . 297 4 Determinants Exist∗ . . . . . . . . . . . . . . . . . . . . . . . . 306 II Geometry of Determinants . . . . . . . . . . . . . . . . . . . . . . 313 1 Determinants as Size Functions . . . . . . . . . . . . . . . . . . 313 III Other Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 1 Laplace’s Expansion∗ . . . . . . . . . . . . . . . . . . . . . . . . 320 Topic: Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Topic: Speed of Calculating Determinants . . . . . . . . . . . . . . . 328 Topic: Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . 331 Chapter Five: Similarity 343 I Complex Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 343 1 Factoring and Complex Numbers; A Review∗ . . . . . . . . . . 344 2 Complex Representations . . . . . . . . . . . . . . . . . . . . . 345 II Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 viii
1 Definition and Examples ... 347 2 Diagonalizability.,·。·.·.·················· 349 3 Eigenvalues and Eigenvectors··· 353 III Nilpotence 361 1 Self-Composition" 361 2 Strings*。364 V Jordan Form..,..··。,。·。············ 375 1 Polynomials of Maps and Matrices◆.··············· 375 2 Jordan Canonical Form*......·....。...· 。。。。。 382 Topic:Method of Powers.························ 395 Topic:Stable Populations.·..·.·.·.·.··.···.····· 399 Topic:Linear Recurrences 401 Appendix A-1 Propositions ........ ·。·.........。...A-1 Quantifiers 4 ····.A-3 Techniques of Proof·.······· 4。 A-5 Sets.Functions,and Relations ... A-7 *Note:starred subsections are optional. 这
1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . 347 2 Diagonalizability . . . . . . . . . . . . . . . . . . . . . . . . . . 349 3 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . 353 III Nilpotence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 1 Self-Composition∗ . . . . . . . . . . . . . . . . . . . . . . . . . 361 2 Strings∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 IV Jordan Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 1 Polynomials of Maps and Matrices∗ . . . . . . . . . . . . . . . . 375 2 Jordan Canonical Form∗ . . . . . . . . . . . . . . . . . . . . . . 382 Topic: Method of Powers . . . . . . . . . . . . . . . . . . . . . . . . . 395 Topic: Stable Populations . . . . . . . . . . . . . . . . . . . . . . . . 399 Topic: Linear Recurrences . . . . . . . . . . . . . . . . . . . . . . . . 401 Appendix A-1 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1 Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3 Techniques of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . A-5 Sets, Functions, and Relations . . . . . . . . . . . . . . . . . . . . . A-7 ∗Note: starred subsections are optional. ix
Chapter One Linear Systems I Solving Linear Systems Systems of linear equations are common in science and mathematics.These two examples from high school science [Onan]give a sense of how they arise. The first example is from Physics.Suppose that we are given three objects, one with a mass known to be 2 kg.and are asked to find the unknown masses. Suppose further that experimentation with a meter stick produces these two balances 一40 —50- 25十 一50 @(c) 2 (②① 15 25州 Since the sum of moments on the left of each balance equals the sum of moments on the right (the moment of an object is its mass times its distance from the balance point),the two balances give this system of two equations. 40h+15c=100 25c=50+50h The second example of a linear system is from Chemistry.We can mix, under controlled conditions,toluene C7Hs and nitric acid HNO3 to produce trinitrotoluene C7H5O6N3 along with the byproduct water(conditions have to be controlled very well,indeed-trinitrotoluene is better known as TNT).In what proportion should those components be mixed?The number of atoms of each element present before the reaction x C7Hs yHNO3 -zC7H506N3 wH2O must equal the number present afterward.Applying that principle to the ele- 1
Chapter One Linear Systems I Solving Linear Systems Systems of linear equations are common in science and mathematics. These two examples from high school science [Onan] give a sense of how they arise. The first example is from Physics. Suppose that we are given three objects, one with a mass known to be 2 kg, and are asked to find the unknown masses. Suppose further that experimentation with a meter stick produces these two balances. c h 2 15 40 50 c 2 h 25 50 25 Since the sum of moments on the left of each balance equals the sum of moments on the right (the moment of an object is its mass times its distance from the balance point), the two balances give this system of two equations. 40h + 15c = 100 25c = 50 + 50h The second example of a linear system is from Chemistry. We can mix, under controlled conditions, toluene C7H8 and nitric acid HNO3 to produce trinitrotoluene C7H5O6N3 along with the byproduct water (conditions have to be controlled very well, indeed— trinitrotoluene is better known as TNT). In what proportion should those components be mixed? The number of atoms of each element present before the reaction x C7H8 + y HNO3 −→ z C7H5O6N3 + w H2O must equal the number present afterward. Applying that principle to the ele- 1