Linear Algebra and Its Applications.pdf 4th Gilbert Strang 带书签

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Linear Algebra and Its Applications.pdf 4th Gilbert Strang 带书签
Contents Preface 1 Matrices and gaussian elimination 1.1 Introduction 1. 2 The Geometry of Linear Equations 1.3 An Example of gaussian elimination 13 1.4 Matrix Notation and Matrix multiplication 1. 5 Triangular Factors and row exchanges ........,.......... 36 1.6 Inverses and Transposes 50 1.7 Spccial Matrices and Applications 66 Review exercises ,,,72 2Ⅴ ector Spaces 2. 1 Vector Spaces and Subspaces 77 2.2 Solving Ax=0 and Ax-b 86 2.3 Linear Independence, Basis, and Dimension ................103 2.4 The Four Fundamental Subspaces 115 2.5 Graphs and Networks ,128 2.6 Linear transformations 140 Review exercises 154 3 Orthogonality 159 3. 1 Orthogonal vectors and Subspaces 159 3.2 Cosines and Projections onto lines 3.3 Projections and least squares .180 3.4 Orthogonal Bases and gram-Schmidt 195 3.5 The Fast Fourier transform 211 Revicw exercises 221 CONTENTS 4 Determinants 224 4.1 Introduction 224 4.2 Properties of the determinant ..226 4.3 Formulas for the determinant 4.4 Applications of Determinants 246 ReⅤ ew exercises .,,256 5 Eigenvalues and eigenvectors 258 5.1 Introduction 258 5.2 Diagonalization of a matrix.......................... 271 5.3 Diffcrencc Equations and Powers Ak 281 5.4 Differential Equations and eat 294 5.5 Complex matrices ..309 5.6 Similarity transformations 323 Review exercises 338 6 Positive definite matrices 342 6.1 Minima. Maxima and saddle points 342 6.2 Tests for Positive definiteness 349 6.3 Singular Value Decomposition.............364 6.4 Minimum Principles 373 6.5 The Finitc Element Mcthod 381 7 Computations with matrices 387 7. 1 Introduction ...,,387 7. 2 Matrix Norm and Condition number 388 7. 3 Computation of eigenvalues 396 7. 4 terative Methods for Ax=b 405 8 Linear Programming and Game Theory 414 8.1 Linear Inequalities 414 8.2 The Simplex Method 419 8.3 The dual problem .,431 8. 4 Network models 441 8.5 Game Theory...,,,,,,,,,,,,,,,,,,,,,,,,,,.,,,448 a Intersection, Sum, and Product of Spaces 456 A.1 The Intersection of Two Vector Spaces 456 A. 2 The Sum of Two Vector Spaces ..457 A. 3 The Cartesian Product of Two Vector Spaces .458 A. 4 The Tensor Product of Two Vector Spaces .458 A. 5 The Kronecker Product A&B of Two matrices 459 CONTENTS b The Jordan Form 463 c Matrix Factorizations 470 D Glossary: A Dictionary for Linear Algebra 472 E MATLAB Teaching Codes 481 F Linear Algebra in a Nutshell 483 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 desirable--to add new problems. Teaching for all these ycars rcquired hundreds of ncw cxam qucstions(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 wasnt a great teacher, but we will give him the benefit of the doubt). Cer tainly the laws of physics are well cxpressed by differential cquations. Newton nccdcd 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 Loward 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. instcad 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 3v+4w. That new vector is in the same plane as v and w. When we take all combinations, we are filling in the whole plane. If i draw v and w on this page, their combinations cv dw fill the page (and beyond), but they don't go up from the page. In the language of linear equations, I can solve cv+dw= b exactly when the vector b lics in the samc planc 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. matrix =2 3 To find combinations of thosc columns, multiply thc matrix by a vcctor(c, d) Linear combinations cv+dw C C2|+d3 34 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 thrcc componcnts to gct right. So w have thrcc cquations to solvc c+ d mcans 2c+3d=5 3c+4a=7 I leave the solution to you. The vector b=(2, 5, 7) does lie in the plane of v and w. If the 7 changes to any other number, then b wont 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=b. 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. Thc 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 hcrc, so you can start thc coursc PREFACE 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,whichiscontinuall updated for the course that is taught every semester. Linear algebra is also on MITs OpenCourseWaresilehttp:/,where18.06becameexceptionalbyincluding videos of the lectures (which you definitely dont have to watch. ). Here is a part of what is available on the web 1. 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 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 0-030-10588-4 The Instructor's Solutions Man- ual 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 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 cntrics. So the coursc 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 ot 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 lincar 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 a can give. Sections 5. 1 and 5.2 are worth careful reading, to see how Ax=nx 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 arc all lincar algebra in disguise, applicd to signal proccssing and partial differential equations and scientific computing(and even GPs). If you look at,youwillseepartofthereasonthatlinearalgebra 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 cxamples to teach what students nccd PREFACE Acknowledgments I enjoyed writing this book, and i certainly hope you enjoy reading it. a big part of the pleasure comes from working with friends. I had wonderful help from Brett Coonley and Cordula robinson and erin maneri. They created the lteX files and drew all the figures. without Brett's 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 LL, U, P]=lu(a)for linear equations Q,R]=gr(a)to make the columns orthogonal s, E]= eig(a)to find cigcnvcctors and cigcnvalucs 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 strung

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