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Table of contents Chapter 1 MATRICES AND GAUSSIAN ELIMINATION 1 1.1 Introduction 1. 2 The Geometry of Linear Equations 3 1.3 An Example of Gaussian Elimination 11 1.4 Matrix Notation and Matrix Multiplication 1.5 Triangular Factors and Row Exchanges 32 1.6 Inverses and Transposes 45 1.7 Special Matrices and applications 8 Review Exercises: Chapter 1 65 Chapter 2 VECTOR SPACEs 69 2.1 Vector Spaces and Subspaces 69 2.2 Solving Ax=0 and Ax=b 77 2.3 Linear Independence. Basis. and Dimension 92 2.4 The Four Fundamental Subspaces 102 2.5 Graphs and Networks 114 2.6 Linear Transformations 125 Review Exercises: Chapter 2 137 it Chapter 3 ORTHOGONALITY 141 3.1 Orthogonal Vectors and Subspaces 141 3.2 Cosines and Projections onto Lines 152 3.3 Projections and Least Squares 160 3.4 Orthogonal Bases and gram-Schmidt 174 3.5 The Fast Fourier transform 188 Review Exercises: Chapter 3 198 Chapter 4 DETERMINANTS 201 4.1 Introduction 201 4.2 Properties of the Determinant 203 4.3 Formulas for the determinant 210 4.4 Applications of Determinants 220 Review Exercises: Chapter 4 230 Table of contents a Chapter 5 EIGENVALUES AND EIGENVECTORS 233 5.1 Introduction 233 5.2 Diagonalization of a Matrix 245 5.4 Differential Equations andeers At 5.3 Difference Equations and Pow 254 266 5.5 Complex Matrices 280 5.6 Similarity Transformations 293 Review Exercises: Chapter 5 307 Chapter 6 POSITIVE DEFINITE MATRICES 311 6.1 Minima. Maxima and Saddle points 311 6.2 Tests for Positive Definiteness 318 6.3 Singular Value Decomposition 331 6.4 Minimum Principles 339 6.5 The Finite Element Method 346 Chapter 7 COMPUTATIONS WITH MATRICES 351 7.1 Introduction 351 7.2 Matrix Norm and Condition Number 352 7. 3 Computation of Eigenvalues 359 7.4 Iterative Methods for Ax=b 367 Chapter 8 LINEAR PROGRAMMING AND GAME THEORY 377 8.1 Linear Inequalities 377 8.2 The Simplex method 382 8.3 The Dual Problem 392 8.4 Network Models 401 8.5 Game Theory 408 Appendix A INTERSECTION, SUM, AND PRODUCT OF SPACES 415 Appendix b THE JORDAN FORM 422 Solutions to selected exercises 428 Matrix factorizations 474 Glossary 476 MATlaB Teaching codes 482 Linear algebra in a Nutshell 488 Preface Revising this textbook has been a special challenge, for a very nice reason. So many people have read this book and taught from it, and even loved it The spirit of the book could never change. This text was written to help our teaching of linear algebra keep up with the enormous importance of this subject--which just continues to grow One step was certainly possible and desirableto add new problems. Teaching for all these years required hundreds of new exam questions(especially with quizzes going onto the web). I think you will approve of the extended choice of problems. The questions are still a mixture of explain and compute--the two complementary approaches to learning this beautiful subject I personally believe that many more people need linear algebra than calculus Isaac Newton might not agree! But he isn't teaching mathematics in the 2lst century (and maybe he wasn't a great teacher, but we will give him the benefit of the doubt) Certainly the laws of physics are well expressed by differential equations. Newton needed calculus--quite right But the scope of science and engineering and management(and life)is now so much wider, and linear algebra has moved into a central place. May I say a little more, because many universities have not yet adjusted the balance toward linear algebra. Working with curved lines and curved surfaces, the first step is always to linearize. Replace the curve by its tangent line, fit the surface by a plane, and the problem becomes linear. The power of this subject comes when you have ten variables or 1000 variables. instead of two You might think I am exaggerating to use the word"beautiful"for a basic course in mathematics. Not at all. This subject begins with two vectors v and w, pointing in different directions. The key step is to take their linear combinations. We multiply to get 3v and 4w, and we add to get the particular combination 30+ 4w. That new vector is in the same plane as v and w. when we take all combinations, we are filling in the hole plane. If I draw v and w on this page, their combinations cu+ dw fill the page (and beyond), but they don't go up from the page In the language of linear equations, I can solve cu +dw= b exactly when the vector b lies in the same plane as v and w Matrices I will keep going a little more to convert combinations of three-dimensional vectors into linear algebra. If the vectors are v=(1, 2, 3)and w =(1, 3, 4), put them into the columns of a matrix 1 matrix Preface To find combinations of those columns, "multiply"the matrix by a vector(c, d) Linear combinations cu+dw 2 c2|+d|3 Those combinations fill a vector space. We call it the column space of the matrix (For these two columns, that space is a plane. To decide if b=(2, 5, 7) is on that plane, we have three components to get right. So we have three equations to solve C+d=2 5 means 2c+3d=5 3c+4d I leave the solution to you. The vector b=(2, 5, 7) does lie in the plane of u and w If the 7 changes to any other number, then b won' t lie in the plane - it will not be a combination of v and w, and the three equations will have no solution Now I can describe the first part of the book, about linear equations Ax The matrix A has n columns and m rows. Linear algebra moves steadily to n vectors in m-dimensional space. We still want combinations of the columns (in the column space). We still get m equations to produce b(one for each row ). Those equations may or may not have a solution. They always have a least-squares solution The interplay of columns and rows is the heart of linear algebra It's not totally easy, but it's not too hard here are four of the central ideas 1. The column space(all combinations of the columns) 2. The row space(all combinations of the rows) 3. The rank(the number of independent columns)(or rows) 4. Elimination(the good way to find the rank of a matrix) i will stop here so you can start the course Web Pages It may be helpful to mention the web pages connected to this book. So many messages come back with suggestions and encouragement, and i hope you will make free use ofeverythingYoucandirectlyaccess,whichiscontinually updated for the course that is taught every semester. Linear algebra is also on MITs OpencourseWaresitehttp.:/ videos of the lectures(which you definitely don' t have to watch.). Here is a part of what is available on the web Lecture schedule and current homeworks and exams with solutions 2. The goals of the course and conceptual questions 3. Interactive Java demos(audio is now included for eigenvalues) 4. Linear Algebra Teaching Codes and MATLAB problems 5. Videos of the complete course(taught in a real classroom) The course page has become a valuable link to the class, and a resource for the students I am very optimistic about the potential for graphics with sound. The bandwidth for Preface voiceover is low, and FlashPlayer is freely available. This offers a quick review(with active experiment), and the full lectures can be downloaded. I hope professors and students worldwide will find these web pages helpful. My goal is to make this book as useful as possible with all the course material I can provide Other Supporting Materials Student Solutions Manual 0-495-01325-0 The Student Solutions Manual provides solutions to the odd-numbered problems in the text Instructor's Solutions Manual 0-030-10568-4 The Instructor's Solutions Manual has teaching notes for each chapter and solutions to all of the problems in the text Structure of the course The two fundamental problems are Ax= b and Ax nx for square matrices a The first problem Ax = b has a solution when A has independent columns. The second problem Ax 2x looks for independent eigenvectors. A crucial part of this course is to learn what independence"means i believe that most of us learn first from examples. You can see that a=1 2 3 does not have independent columns Column 1 plus column 2 equals column 3. A wonderful theorem of linear algebra says that the three rows are not independent either. The third row must lie in the same plane as the first two rows. Some combination of rows I and 2 will produce row 3. You might find that combination quickly (i didnt). In the end i had to use elimination to discover that the right combination uses 2 times row 2, minus row 1 Elimination is the simple and natural way to understand a matrix by producing a lot of zero entries. So the course starts there But dont stay there too long! You have to get from combinations of the rows, to independence of the rows, to"dimension of the row space. That is a key goal, to see whole spaces of vectors: the row space and the column space and the nullspace A further goal is to understand how the matrix acts. When A multiplies x it produces the new vector Ax. The whole space of vectors moves- it is"transformed 'by A special transformations come from particular matrices, and those are the foundation stones of linear algebra: diagonal matrices, orthogonal matrices, triangular matrices, symmetric matrices The eigenvalues of those matrices are special too. I think 2 by 2 matrices provide terrific examples of the information that eigenvalues 2 can give. Sections 5. 1 and 5.2 are worth careful reading to see how Ax =hx is useful. here is a case in which small matrices allow tremendous insight Overall, the beauty of linear algebra is seen in so many different ways 1. Visualization. Combinations of vectors Spaces of vectors. Rotation and reflection and projection of vectors. Perpendicular vectors. Four fundamental subspaces 2. Abstraction: Independence of vectors. Basis and dimension of a vector space Linear transformations. Singular value decomposition and the best basis 3. Computation. Elimination to produce zero entries. Gram-Schmidt to produce orthogonal vectors. Eigenvalues to solve differential and difference equations 4. Applications. Least-squares solution when Ax= b has too many equations. Dif- ference equations approximating differential equations. Markov probability matrices (the basis for Google!). Orthogonal eigenvectors as principal axes(and more.) To go further with those applications, may I mention the books published by Wellesley- Cambridge Press. They are all linear algebra in disguise, applied to signal processing and partial differential equations and scientific computing(and even GPS). If you look athttp://ww.wellesleycambridge.comyouwillseepartofthereasonthatlinearalgebra is so widely used After this preface, the book will speak for itself. You will see the spirit right away The emphasis is on understanding try to explain rather than to deduce. This is a book about real mathematics, not endless drill. In class, i am constantly working with examples to teach what students need Acknowledgments I enjoyed writing this book, and I certainly hope you enjoy reading it. a big part of the leasure comes from working with friends i had wonderful help from brett Coonle and Cordula robinson and Erin Maneri. They created the iteX files and drew all the figures. Without Bretts steady support I would never have completed this new edition Earlier help with the Teaching Codes came from Steven Lee and Cleve moler Those follow the steps described in the book; MATLAB and Maple and Mathematica are faster for large matrices. All can be used (optionally) in this course. I could have added Factorization" to that list above, as a fifth avenue to the understanding of matrices [L, U, P]= lu(a) for linear equations Q, R]= qr(a) to make the columns orthogonal S,E]= eig (a)to find eigenvectors and eigenvalues In giving thanks, I never forget the first dedication of this textbook, years ago That was a special chance to thank my parents for so many unselfish gifts. Their example is an inspiration for my life And I thank the reader too, hoping you like this book Gilbert strang Chapter Matrices and Gaussian elimination 1.1 NTRODUCTION This book begins with the central problem of linear algebra: solving linear equations The most important case, and the simplest, is when the number of unknowns equals the number of equations. We have n equations in n unknowns, starting with n=2: wo equations 1x+2y=3 Two unknowns 4x 5 The unknowns are x and y. I want to describe two ways, elimination and determinants, to solve these equations. Certainly x and y are determined by the numbers 1, 2, 3, 4, 5,6 The question is how to use those six numbers to solve the system. 1. Elimination Subtract 4 times the first equation from the second equation. This eliminates x from the second equation, and it leaves one equation for y (equation 2)-4(equation 1) (2) Immediately we know y= 2. Then x comes from the first equation lx 2y =3 Back-substitutic 1x+2(2)=3 gives x=-1. (3) Proceeding carefully, we check that x and y also solve the second equation. This should work and it does: 4 times 1)plus 5 times (y= 2)equals 6 2. Determinants The solution y 2 depends completely on those six numbers in the equations. There must be a formula for y(and also x). It is a"ratio of determinants and I hope you will allow me to write it down directly 46 12 45 2 Chapter 1 Matrices and Gaussian Elimination That could seem a little mysterious, unless you already know about 2 by 2 determinants They gave the same answer y 2, coming from the same ratio of -6 to-3 If we stay with determinants(which we dont plan to do), there will be a similar formula to compute the other unknown,x 3.5-2.6 (5) et me compare those two approaches, looking ahead to real problems when n is much larger(n=1000 is a very moderate size in scientific computing). The truth is that direct use of the determinant formula for 1000 equations would be a total disaster It would use the million numbers on the left sides correctly, but not efficiently. We will find that formula( Cramer's Rule) in Chapter 4, but we want a good method to solve 1000 equations in Chapter 1 That good method is Gaussian Elimination. This is the algorithm that is constantly used to solve large systems of equations From the examples in a textbook (n=3 close to the upper limit on the patience of the author and reader) you might not see much difference. Equations(2)and(4)used essentially the same steps to find y= 2. Certainly x came faster by the back-substitution in equation 3)than the ratio in(5). For larger n there is absolutely no question. Elimination wins(and this is even the best way to compute determinants) The idea of elimination is deceptively simple-you will master it after a few exam les. It will become the basis for half of this book simplifying a matrix so that we can understand it. Together with the mechanics of the algorithm, we want to explain four deeper aspects in this chapter. They are 1. Linear equations lead to geometry of planes. It is not easy to visualize a nine- dimensional plane in ten-dimensional space. It is harder to see ten of those planes intersecting at the solution to ten equations--but somehow this is almost possible. Our example has two lines in Figure 1. 1, meeting at the point(x, y)=(1, 2) Linear algebra moves that picture into ten dimensions, where the intuition has to imagine the geometry (and gets it right) 2. We move to matrix notation, writing the n unknowns as a vector x and the n equations as Ax =b. We multiply a byelimination matrices"to reach an up per triangular matrix U. Those steps factor A into L times U, where L is lower x+2y=3 x+2y=3 4x+5y=6 4x+8 4x+8y=12 One solution(x, y)=(1, 2) Parallel: No solution Whole line of solutions Figure 1.1 The example has one solution Singular cases have none or too many

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