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Linear AlgebraJim Hefferon21131 23 121x1·13x1· 1 2x1· 3 121686 28 1 NotationR, R+, Rnreal numbers, reals greater than 0, n-tuples of realsN 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 mattersV, W, U vector spacesv, w vectors0,0Vzero vector, zero vector of VB, D basesEn= e1, . . . , en standard basis for Rnβ,δ basis vectorsRepB(v) matrix representing the vectorPnset of n-th degree polynomialsMn×mset of n×m matrices[S] span of the set SM ⊕ N direct sum of subspacesV∼=W isomorphic spacesh, g homomorphisms, linear mapsH, G matricest, s transformations; maps from a space to itselfT, S square matricesRepB,D(h) matrix representing the map hhi,jmatrix entry from row i, column jZn×m, Z, In×n, I zero matrix, identity matrix|T| determinant of the matrix TR(h), N (h) rangespace and nullspace of the map hR∞(h), N∞(h) generalized rangespace and nullspaceLower case Greek alphabetname character name character name characteralpha α iota ι rho ρbeta β kappa κ sigma σgamma γ lambda λ tau τdelta δ mu µ upsilon υepsilon nu ν phi φzeta ζ xi ξ chi χeta η omicron o psi ψtheta θ pi π omega ωCover. This is Cramer’s Rule for the system x1+ 2x2= 6, 3x1+ x2= 8. The size ofthe first box is the determinant shown (the absolute value of the size is the area). Thesize of the second box is x1times that, and equals the size of the final box. Hence, x1is the final determinant divided by the first determinant. PrefaceThis book helps students to master the material of a standard US undergraduatelinear algebra course.The material is standard in that the topics covered are Gaussian reduction,vector spaces, linear maps, determinants, and eigenvalues and eigenvectors. An-other standard is book’s audience: sophomores or juniors, usually with a back-ground of at least one semester of calculus. The help that it gives to studentscomes from taking a developmental approach — this book’s presentation empha-sizes motivation and naturalness, driven home by a wide variety of examples andby extensive and careful exercises.The developmental approach is the feature that most recommends this bookso I will say more. Courses in the beginning of most mathematics programsfocus less on understanding theory and more on correctly applying formulasand algorithms. Later courses ask for mathematical maturity: the ability tofollow different types of arguments, a familiarity with the themes that underliemany mathematical investigations such as elementary set and function facts,and a capacity for some independent reading and thinking. Linear algebra isan ideal spot to work on the transition. It comes early in a program so thatprogress made here pays off later, but also comes late enough that students areserious about mathematics, often majors and minors. The material is accessible,coherent, and elegant. There are a variety of argument styles, including proofsby contradiction, if and only if statements, and proofs by induction. And,examples are plentiful.Helping readers start the transition to being serious students of the subject ofmathematics itself means taking the mathematics seriously, so all of the resultsin this book are proved. On the other hand, we cannot assume that studentshave already arrived and so in contrast with more abstract texts, we give manyexamples and they are often quite detailed.Some linear algebra books begin with extensive computations of linear sys-tems, matrix multiplications, and determinants. Then, when the concepts —vector spaces and linear maps — finally appear, and definitions and proofs start,often the abrupt change brings students to a stop. In this book, while we startwith a computational topic, linear reduction, from the first we do more thancompute. We do linear systems quickly but completely, including the proofsneeded to justify what we are computing. Then, with the linear systems workas motivation and at a point where the study of linear combinations seems nat-ural, the second chapter starts with the definition of a real vector space. In theiii schedule below, this occurs by the end of the third week.Another example of our emphasis on motivation and naturalness is that thethird chapter on linear maps does not begin with the definition of homomor-phism, but with isomorphism. The definition of isomorphism is easily motivatedby the observation that some spaces are “just like” others. After that, the nextsection takes the reasonable step of defining homomorphism by isolating theoperation-preservation idea. This approach loses mathematical slickness, but itis a good trade because it gives to students a large gain in sensibility.One aim of our developmental approach is to present the material in such away that students can see how the ideas arise, and perhaps can picture them-selves doing the same type of work.The clearest example of the developmental approach is the exercises. A stu-dent progresses most while doing the exercises, so the ones included here havebeen selected with great care. Each problem set ranges from simple checks toreasonably involved proofs. Since an instructor usually assigns about a dozen ex-ercises after each lecture, each section ends with about twice that many, therebyproviding a selection. There are even a few problems that are challenging puz-zles taken from various journals, competitions, or problems collections. (Theseare marked with a ‘?’ and as part of the fun, the original wording has beenretained as much as possible.) In total, the exercises are aimed to both buildan ability at, and help students experience the pleasure of, doing mathematics.Applications and computers. The point of view taken here, that studentsshould think of linear algebra as about vector spaces and linear maps, is nottaken to the complete exclusion of others. Applications and computing areimportant and vital aspects of the subject. Consequently, each of this book’schapters closes with a few application or computer-related topics. Some are: net-work flows, the speed and accuracy of computer linear reductions, Leontief In-put/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 asindependent projects. Most simply give a reader a taste of the subject, discusshow linear algebra comes in, point to some further reading, and give a fewexercises. In short, these topics invite readers to see for themselves that linearalgebra is a tool that a professional must have.The license. This book is freely available. You can download and read itwithout restriction. Class instructors can print copies for students and chargefor those. See http://joshua.smcvt.edu/linearalgebra for more license in-formation.That page also contains the latest version of this book, and the latest versionof the worked answers to every exercise. Also there, I provide the LATEX sourceof the text and some instructors may wish to add their own material. If youlike, you can send such additions to me and I may possibly incorporate theminto future editions.I am very glad for bug reports. I save them and periodically issue updates;people who contribute in this way are acknowledged in the text’s source files.iv For people reading this book on their own. This book’s emphasis onmotivation and development make it a good choice for self-study. But while aprofessional instructor can judge what pace and topics suit a class, if you arean independent student then you may find some advice helpful.Here are two timetables for a semester. The first focuses on core material.week Monday Wednesday Friday1 One.I.1 One.I.1, 2 One.I.2, 32 One.I.3 One.II.1 One.II.23 One.III.1, 2 One.III.2 Two.I.14 Two.I.2 Two.II Two.III.15 Two.III.1, 2 Two.III.2 exam6 Two.III.2, 3 Two.III.3 Three.I.17 Three.I.2 Three.II.1 Three.II.28 Three.II.2 Three.II.2 Three.III.19 Three.III.1 Three.III.2 Three.IV.1, 210 Three.IV.2, 3, 4 Three.IV.4 exam11 Three.IV.4, Three.V.1 Three.V.1, 2 Four.I.1, 212 Four.I.3 Four.II Four.II13 Four.III.1 Five.I Five.II.114 Five.II.2 Five.II.3 reviewThe second timetable is more ambitious. It supposes that you know One.II, theelements of vectors, usually covered in third semester calculus.week Monday Wednesday Friday1 One.I.1 One.I.2 One.I.32 One.I.3 One.III.1, 2 One.III.23 Two.I.1 Two.I.2 Two.II4 Two.III.1 Two.III.2 Two.III.35 Two.III.4 Three.I.1 exam6 Three.I.2 Three.II.1 Three.II.27 Three.III.1 Three.III.2 Three.IV.1, 28 Three.IV.2 Three.IV.3 Three.IV.49 Three.V.1 Three.V.2 Three.VI.110 Three.VI.2 Four.I.1 exam11 Four.I.2 Four.I.3 Four.I.412 Four.II Four.II, Four.III.1 Four.III.2, 313 Five.II.1, 2 Five.II.3 Five.III.114 Five.III.2 Five.IV.1, 2 Five.IV.2In the table of contents I have marked subsections as optional if some instructorswill pass over them in favor of spending more time elsewhere.You might pick one or two topics that appeal to you from the end of eachchapter. You’ll get more from these if you have access to computer softwarethat can do any big calculations. I recommend Sage, freely available fromhttp://sagemath.org.v My main advice is: do many exercises. I have marked a good sample with’s in the margin. For all of them, you must justify your answer either with acomputation or with a proof. Be aware that few inexperienced people can writecorrect proofs. Try to find someone with training to work with you on this.Finally, if I may, a caution for all students, independent or not: I cannotoveremphasize how much the statement that I sometimes hear, “I understandthe material, but it’s only that I have trouble with the problems” is mistaken.Being able to do things with the ideas is their entire point. The quotes belowexpress this sentiment admirably. They state what I believe is the key to boththe beauty and the power of mathematics and the sciences in general, and oflinear algebra in particular; I took the liberty of formatting them as verse.I know of no better tacticthan the illustration of exciting principlesby well-chosen particulars.–Stephen Jay GouldIf you really wish to learnthen you must mount the machineand become acquainted with its tricksby actual trial.–Wilbur WrightJim HefferonMathematics, Saint Michael’s CollegeColchester, Vermont USA 05439http://joshua.smcvt.edu2011-Jan-01Author’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 forsome reason texts have traditionally not given attributions for questions. I havechanged that here where I was sure of the source. I would be glad to hear fromanyone who can help me to correctly attribute others of the questions.vi ContentsChapter One: Linear Systems 1I Solving Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . 11 Gauss’ Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Describing the Solution Set . . . . . . . . . . . . . . . . . . . . 113 General = Particular + Homogeneous . . . . . . . . . . . . . . 20II Linear Geometry of n-Space . . . . . . . . . . . . . . . . . . . . . 321 Vectors in Space . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Length and Angle Measures∗. . . . . . . . . . . . . . . . . . . 39III Reduced Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . 461 Gauss-Jordan Reduction . . . . . . . . . . . . . . . . . . . . . . 462 Row Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . 52Topic: Computer Algebra Systems . . . . . . . . . . . . . . . . . . . 61Topic: Input-Output Analysis . . . . . . . . . . . . . . . . . . . . . . 63Topic: Accuracy of Computations . . . . . . . . . . . . . . . . . . . . 67Topic: Analyzing Networks . . . . . . . . . . . . . . . . . . . . . . . . 71Chapter Two: Vector Spaces 77I Definition of Vector Space . . . . . . . . . . . . . . . . . . . . . . 781 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . 782 Subspaces and Spanning Sets . . . . . . . . . . . . . . . . . . . 89II Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . 991 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . 99III Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 1101 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163 Vector Spaces and Linear Systems . . . . . . . . . . . . . . . . 1224 Combining Subspaces∗. . . . . . . . . . . . . . . . . . . . . . . 129Topic: Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138Topic: Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Topic: Voting Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . 144Topic: Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . 150vii Chapter Three: Maps Between Spaces 157I Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1571 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . 1572 Dimension Characterizes Isomorphism . . . . . . . . . . . . . . 166II Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 1741 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1742 Rangespace and Nullspace . . . . . . . . . . . . . . . . . . . . . 181III Computing Linear Maps . . . . . . . . . . . . . . . . . . . . . . . 1931 Representing Linear Maps with Matrices . . . . . . . . . . . . . 1932 Any Matrix Represents a Linear Map∗. . . . . . . . . . . . . . 203IV Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 2101 Sums and Scalar Products . . . . . . . . . . . . . . . . . . . . . 2102 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . 2133 Mechanics of Matrix Multiplication . . . . . . . . . . . . . . . . 2204 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229V Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2361 Changing Representations of Vectors . . . . . . . . . . . . . . . 2362 Changing Map Representations . . . . . . . . . . . . . . . . . . 240VI Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2481 Orthogonal Projection Into a Line∗. . . . . . . . . . . . . . . . 2482 Gram-Schmidt Orthogonalization∗. . . . . . . . . . . . . . . . 2523 Projection Into a Subspace∗. . . . . . . . . . . . . . . . . . . . 258Topic: Line of Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . 267Topic: Geometry of Linear Maps . . . . . . . . . . . . . . . . . . . . 272Topic: Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . 279Topic: Orthonormal Matrices . . . . . . . . . . . . . . . . . . . . . . 285Chapter Four: Determinants 291I Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2921 Exploration∗. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2922 Properties of Determinants . . . . . . . . . . . . . . . . . . . . 2973 The Permutation Expansion . . . . . . . . . . . . . . . . . . . . 3014 Determinants Exist∗. . . . . . . . . . . . . . . . . . . . . . . . 309II Geometry of Determinants . . . . . . . . . . . . . . . . . . . . . . 3171 Determinants as Size Functions . . . . . . . . . . . . . . . . . . 317III Other Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3241 Laplace’s Expansion∗. . . . . . . . . . . . . . . . . . . . . . . . 324Topic: Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 329Topic: Speed of Calculating Determinants . . . . . . . . . . . . . . . 332Topic: Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . 335Chapter Five: Similarity 347I Complex Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 3471 Factoring and Complex Numbers; A Review∗. . . . . . . . . . 3482 Complex Representations . . . . . . . . . . . . . . . . . . . . . 349II Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351viii 1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . 3512 Diagonalizability . . . . . . . . . . . . . . . . . . . . . . . . . . 3533 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . 357III Nilpotence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3651 Self-Composition∗. . . . . . . . . . . . . . . . . . . . . . . . . 3652 Strings∗. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368IV Jordan Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3791 Polynomials of Maps and Matrices∗. . . . . . . . . . . . . . . . 3792 Jordan Canonical Form∗. . . . . . . . . . . . . . . . . . . . . . 386Topic: Method of Powers . . . . . . . . . . . . . . . . . . . . . . . . . 399Topic: Stable Populations . . . . . . . . . . . . . . . . . . . . . . . . 403Topic: Linear Recurrences . . . . . . . . . . . . . . . . . . . . . . . . 405Appendix A-1Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3Techniques of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . A-5Sets, Functions, and Relations . . . . . . . . . . . . . . . . . . . . . A-7∗Note: starred subsections are optional.ix [...]... describe solution sets as above, with a particular solution vector added to an unrestricted linear combination of some other vec- tors? The solution sets we described with unrestricted parameters were easily seen to have infinitely many solutions so an answer to this question could tell us something about the size of solution sets. An answer to that question could also help us picture the solution sets,... a non- 0 solution, and thus by the prior paragraph has infinitely many solutions). QED This table summarizes the factors affecting the size of a general solution. number of solutions of the associated homogeneous system particular solution exists? one infinitely many yes unique solution infinitely many solutions no no solutions no solutions The factor on the top of the table is the simpler one. When we... no solution has a solution set that is empty. In these cases the solution set is easy to describe. Solution sets are a challenge to describe only when they contain many elements. 20 Chapter One. Linear Systems (b) a two-parameter solution set; (c) a three-parameter solution set. ? 2.30 (a) Solve the system of equations. ax + y = a 2 x + ay = 1 For what values of a does the system fail to have solutions,... whether a system has a solution then you must justify a yes response by producing the solution and must justify a no response by showing that no solution exists. Exercises 1.17 Use Gauss’ method to find the unique solution for each system. (a) 2x + 3y = 13 x − y = −1 (b) x − z = 0 3x + y = 1 −x + y + z = 4 1.18 Use Gauss’ method to solve each system or conclude ‘many solutions’ or ‘no solutions’. (a) 2x... concluding a system has a solution? 1.20 For which values of k are there no solutions, many solutions, or a unique solution to this system? x − y = 1 3x − 3y = k 1.21 This system is not linear, in some sense, 2 sin α − cos β + 3 tan γ = 3 4 sin α + 2 cos β − 2 tan γ = 10 6 sin α − 3 cos β + tan γ = 9 Section I. Solving Linear Systems 11 1.31 Is there a two-unknowns linear system whose solution set is all... all so clear when you explain it!” said the Poor Nut. “Do you mean like 6x + 9y = 12 and 15x + 18y = 21?” “Quite so, ” said the Great Mathematician, pulling out his bassoon. “Indeed, the system has a unique solution. Can you find it?” “Good heavens!” cried the Poor Nut, “I am baffled.” Are you? [Am. Math. Mon., Jan. 1963] I.2 Describing the Solution Set A linear system with a unique solution has a solution... seen to be non- 0, and so sv = v. Now, apply Lemma 3.8 to conclude that a solution set {p + h h solves the associated homogeneous system} is either empty (if there is no particular solution p), or has one element (if there is a p and the homogeneous system has the unique solution 0), or is infinite (if there is a p and the homogeneous system has a non- 0 solution, and thus by the prior... an infinite solution set conforming to the pattern. We can think of the other two kinds of solution sets as fitting the same pattern. A one-element solution set fits the pattern in that it has a particular solution, and Section I. Solving Linear Systems 13 top equation, substituting for y in the first equation x + (−1 + z − w) + z − w = 1 and solving for x yields x = 2 − 2z + 2w. Thus, the solution set... theorem states, and as discussed at the start of this subsection, in this single-solution case the general solution results from taking the particular solu- tion and adding to it the unique solution of the associated homogeneous system. 3.10 Example Also discussed at the start of this subsection is that the case where the general solution set is empty fits the ‘General = Particular + Homogeneous’ pattern.... v start finish Section I. Solving Linear Systems 27 3.11 Corollary Solution sets of linear systems are either empty, have one element, or have infinitely many elements. Proof. We’ve seen examples of all three happening so we need only prove that those are the only possibilities. First, notice a homogeneous system with at least one non- 0 solution v has infinitely many solutions because the set of . pair (−1, 5) is a solution of this system.3x1+ 2x2= 7−x1+ x2= 6In contrast, (5,−1) is not a solution.Finding the set of all solutions is solving the system.. x1= 3 and so the system has a uniquesolution: the solution set is { (3, 1, 3)}.Most of this subsection and the next one consists of examples of solvinglinear