线性代数导论gilbert_strang:第四版 英文版

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资源内容是gilbert_strang呕心沥血之作线性代数导论第四版,是英文版本的
Table of Contents 1 Introduction to Vectors 1. 1 Vectors and Linear Combinations 2 Lengths and Dot Products 1. 3 Matrices 22 2 Solving Linear Equations 31 2. 1 Vectors and Linear Equations 2.2 The Idea of elimination 45 2.3 Elimination Using Matrices 57 2.4 Rules for Matrix Operations 68 2.5 Inverse matrices 82 2.6 Elimination= Factorization :A s LU 96 2.7 Transposes and permutations 108 3 Vector Spaces and Subspaces 121 3.1 Spaces of vectors 121 3. 2 The Nullspace of A: Solving Ax =0 133 3.3 The rank and the row reduced form 145 3.4 The Complete Solution to Ax b 156 3.5 Independence Basis and dimension ········4吾··· 169 3.6 Dimensions of the Four Subspaces 185 4 orthogonality 196 4.1 Orthogonality of the Four Subspaces .,,,,,,,,,,,,.,196 4.2 Projections 207 4.3 Least Squares Approximations 219 4. 4 Orthogonal Bases and gram-Schmidt 231 Determinants 245 5.1 The Properties of Determinants ..245 5.2 Permutations and Cofactors 256 5.3 Cramer's rule Inverses and volumes 270 Table of contents 6 Eigenvalues and Eigenvectors 284 6.1 Intr ion to Eigenvalues 284 6.2 Diagonalizing a matrix 299 6.3 Applications to Differential Equations 313 6. 4 Symmetric Matrices ······ 331 6.5 Positive Definite Matrices 343 6.6 Similar matrices 356 6.7 Singular Value Decomposition(SVD) 364 7 Linear Transformations 376 7. 1 The Idea of a Linear transformation 376 7. 2 The Matrix of a Linear transformation 385 7. 3 Diagonalization and the Pseudoinverse 400 8 Applications 4I0 8.1 Matrices in Engineering 410 8.2 Graphs and Networks..,......... 421 8.3 Markov Matrices, Population, and Economics 432 8.4 Linear Programming 441 8.5 Fouricr Series: Linear Algebra for Functions 448 8.6 Linear Algebra for Statistics and Probability 454 8.7 Computer Graphics 460 9 Numerical Linear Algebra 466 9.1 Gaussian elimination in Practice 466 9.2 Norms and Condition Numbers 476 9.3 Iterative Methods and Preconditioners 482 10 Complex Vectors and Matrices 494 10.1 Complex Numbers ......... 494 10.2 Hermitian and Unitary Matrices 502 10.3 The Fast Fourier Transform 510 Solutions to selected Exercises 5I7 Conceptual Questions for Review 553 Glossary: A Dictionary for Linear Algebra 558 Matrix Factorizations 565 Teaching codes 567 Index 568 Linear Algebra in a Nutshell 575 Preface I will be happy with this preface if three important points come through clearly 1. The beauty and variety of linear algebra, and its extreme usefulness 2. The goals of this book and the new features in this Fourth edition 3. The steady support from our linear algebra websites and the video lectures May i begin with notes about two websites that are constantly used, and the new one ocw.mit.edu Messages come from thousands of students and faculty about linear algebra on this Open Course Ware site. The 18.06 course includes video lectures of a complete semester of classes. Those lectures offer an independent review of the whole subject based on this textbook--the professor's time stays free and the student's time can be 3 a.m (The reader doesn t have to be in a class at all. ) a million viewers around the world have seen these videos(amazing ). I hope you find them helpful web. mit. edu/18.06 This site has homeworks and exams(with solutions )for the current as useful as possible, with all the course material we can provide soat. eestions, Java demos, course as it is taught, and as far back as 1996. There are also review ql Teaching Codes, and short essays (and the video lectures). My go is to make this book math. mit. edu/linearalgebra The newest website is devoted specifically to this Fourth edi tion. It will be a permanent record of ideas and codes and good problems and solutions Several sections of the book are directly available online, plus notes on teaching linear lgebra. The content is growing quickly and contributions are welcome from everyone The fourth edition Thousands of readers know earlier editions of Introduction to Linear algebra. The new cover shows the Four Fundamental Subspaces-the row space and nullspace are on the left side, the column space and the nullspace of A are on the right. It is not usual to put the central ideas of the subject on display like this! You will meet those four spaces in Chapter 3, and you will understand why that picture is so central to linear algebra Those were named the Four Fundamental Subspaces in my first book, and they start from a matrix A. Each row of A is a vector in n-dimensional space. When the matrix Preface has m rows, each column is a vector in m-dimensional space. The crucial operation in linear algebra is taking linear combinations of vectors.(That idea starts on page 1 of the book and never stops. ) When we take all linear combinations of the column vectors, we get the column space. If this space includes the vector b, we can solve the equation Ax = b I have to stop here or you wont read the book. May i call special attention to the new Section 1.3 in which these ideas come early-with two specific examples. You are not expected to catch every detail of vector spaces in one day! But you will see the first matrices in the book, and a picture of their column spaces, and even an inverse matrix. You will be learning the language of linear algebra in the best and most efficient way: by using it Every section of the basic course now ends with Challenge Problems. They follow a large collection of review problems, which ask you to use the ideas in that section-the dimension of the column space, a basis for that space, the rank and inverse and determinant and eigenvalues of A. Many problems look for computations by hand on a small matrix and they have been highly praised. The new Challenge Problems go a step further, and sometimes they go deeper. Let me give four examples Section 2.1: Which row exchanges of a Sudoku matrix produce another Sudoku matrix? Section 2.4 From the shapes of A, B, C, is it faster to compute AB times C or A times BC? Background: The great fact about multiplying matrices is that AB timesc gives the same answer as A times BC. This simple statement is the reason behind the rule for matrix multiplication. If AB is square and C is a vector, it's faster to do bc first. Then multiply by A to produce ABC. The question asks about other shapes of A, B, and C Section 3. 4: If Ax b and Cx= b have the same solutions for every b, is A = C? Section 4.1: what conditions on the four vectors r, n,c,e allow them to be bases for the row space, the nullspace, the column space, and the left nullspace of a 2 by 2 matrix? The Start of the Course The equation Ax b uses the language of linear combinations right away. The vector Ax is a combination of the columns of A. The equation is asking for a combination that produces b. The solution vector x comes at three levels and all are important 1. Direct solution to find x by forward elimination and back substitution 2. Matrix solution using the inverse of A: x= A b(if A has an inverse). 3. Vector space solution x y +z as shown on the cover of the book Particular solution(to Ay b)plus nullspace solution(to Az=0) Direct elimination is the most frequently used algorithm in scientific computing, and the idea is not hard. Simplify the matrix A so it becomes triangular--then all solutions come quickly. I dont spend forever on practicing elimination, it will get learned The speed of every new supercomputer is tested on Ax 6: it's pure linear algebra IBM and Los Alamos announced a new world record of 10 5 operations per second in 2008 Preface That petaflop speed was reached by solving many equations in parallel. High performance computers avoid operating on single numbers, they feed on whole submatrices The processors in the Roadrunner are based on the Cell Engine in PlayStation 3 What can I say, video games are now the largest market for the fastest computations Even a supercomputer doesnt want the inverse matrix: too slow. Inverses give the sim- plest formula= A b but not the top speed. And everyone must know that determinants are even slower-there is no way a linear algebra course should begin with formulas for the determinant of an n by n matrix. Those formulas have a place, but not first place Structure of the Textbook Already in this preface, you can see the style of the book and its goal. That goal is serious, to explain this beautiful and useful part of mathematics. You will see how the applications of linear algebra reinforce the key ideas. I hope every teacher will learn something new; amiliar ideas can be seen in a new way. The book moves gradually and steadily from numbers to vectors to subspaces--each level comes naturally and everyone can get it Here are ten points about the organization of this book: 1. Chapter 1 starts with vectors and dot products. If the class has met them before, focus quickly on linear combinations. The new Section 1.3 provides three indepen dent vectors whose combinations fill all of 3-dimensional space, and three depen dent vectors in a plane. Those two examples are the beginning of linear algebra. 2, Chapter 2 shows the row picture and the column picture of Ax =b. The heart of inear algebra is in that connection between the rows of A and the columns: the same numbers but very different pictures. Then begins the algebra of matrices: an elimination matrix E multiplies A to produce a zero. The goal here is to capture the whole process start with A and end with an upper triangular U Elimination is seen in the beautiful form A= LU. The lower triangular L holds all the forward elimination steps, and u is the matrix for back substitution 3. Chapter 3 is linear algebra at the best level: subspaces. The column space contains all linear combinations of the columns. The crucial question is: How many of those columns are needed? The answer tells us the dimension of the column space, and the key information about A. We reach the Fundamental Theorem of Linear Algebra 4. Chapter 4 has m equations and only n unknowns. It is almost sure that Ax= b has no solution. We cannot throw out equations that are close but not perfectly exact When we solve by least squares, the key will be the matrix ATA. This wonderful matrix AA appears everywhere in applied mathematics, when A is rectangular 5. Determinants in Chapter 5 give formulas for all that has come before-inverses pivots, volumes in n-dimensional space, and more. We dont need those formulas to compute! They slow us down. But det A=0 tells when a matrix is singular, and that test is the key to eigenvalues retrace 6. Section 6. 1 introduces eigenvaLues for 2 by 2 matrices. Many courses want to see eigenvalues early. It is completely reasonable to come here directly from Chapter because the determinant is easy for a 2 by 2 matrix. The key equation is Ax =lr Eigenvalues and eigenvectors are an astonishing way to understand a square matrix They are not for Ax = b, they are for dynamic equations like du /dt = Au The idea is always the same: follow the eigenvectors. In those special directions A acts like a single number(the eigenvalue n)and the problem is one-dimensional Chapter 6 is full of applications. One highlight is diagonalizing a symmetric matrix. Another highlight--not so well known but more important every dayis the diagonalization of any matrix. This needs two sets of eigenvectors, not one, and they come(of course! ) from A A and AA. This Singular Value Decomposition often marks the end of the basic course and the start of a second course 7. Chapter 7 explains the linear transformation approach--it is linear algebra without coordinates, the ideas without computations. Chapter 9 is the oppositeall about how Ax= b and Ax x are really solved. Then Chapter 10 moves from real numbers and vectors to complex vectors and matrices. The Fourier matrix F is the most important complex matrix we will ever see. And the Fast Fourier Transform (multiplying quickly by F and F-l)is a revolutionary algorithm 8. Chapter 8 is full of applications, more than any single course could need 8.1 Matrices in Engineering-differential equations replaced by matrix equations 8.2 Graphs and Networks-leading to the edge-node matrix for Kirchhoff's Laws 8.3 Markov Matrices-as in Google's Pagerank algorithm 8.4 Linear programming-a new requirement>0 and minimization of the cost 8.5 Fourier Series-linear algebra for functions and digital signal processing 8.6 Matrices in Statistics and Probability-Ax= b is weighted by average errors 8.7 Computer Graphics-matrices move and rotate and compress images 9. Every section in the basic course ends with a Review of the Key Ideas 10. How should computing be included in a linear algebra course? It can open a new understanding of matrices-every class will find a balance. I chose the language of MATLAB as a direct way to describe linear algebra: eig(ones(4) will produce the eigenvalues 4, 0, 0, 0 of the 4 by 4 all-ones matrix, Go to netlib. org for codes You can freely choose a different system. More and more software is open source The new website math. mit. edullinearalgebra provides further ideas about teaching and learning. Please contribute! Good problems are welcome by email: gs@math. mit. edu Send new applications too, linear algebra is an incredibly useful subject Preface XX The variety of Linear Algebra Calculus is mostly about one special operation(the derivative)and its inverse(the integral) Of course I admit that calculus could be important.. But so many applications of math- ematics are discrete rather than continuous, digital rather than analog. The century of data has begun! You will find a light-hearted essay called"Too Much Calculus"on my website The truth is that vectors and matrices have become the language to know Part of that language is the wonderful variety of matrices. Let me give three examples: Symmetric matrix Orthogonal matrix Triangular matrix 2-10 11 0 0012 111 001 11 0001 a key goal is learning to"reada matrix. You need to see the meaning in the numbers This is really the essence of mathematics-patterns and their meaning May i end with this thought for professors. You might feel that the direction is right, and wonder if your students are ready. Just give them a chance! Literally thousands of students have written to me, frequently with suggestions and surprisingly often with thanks. They know this course has a purpose because the professor and the book are on their side Linear algebra is a fantastic subject enjoy it Help with This Book I cant even name all the friends who helped me, beyond thanking Brett Coonley at MIt and valutone in Mumbai and siaM in philadelphia for years of constant and dedicated support. The greatest encouragement of all is the feeling that you are doing something worthwhile with your life. Hundreds of generous readers have sent ideas and examples and corrections(and favorite matrices! that appear in this book. Thank you all Background of the author This is my eighth textbook on linear algebra, and I have not written about myself before I hesitate to do it now. It is the mathematics that is important, and the reader. The next paragraphs add something personal as a way to say that textbooks are written by people I was born in Chicago and went to school in Washington and cincinnati and St louis My college was MIT (and my linear algebra course was extremely abstract). After that came Oxford and UCLA, then back to MIT for a very long time. I don' t know how many thousands of students have taken 18.06(more than a million when you include the videos on ocw. mit. edu). The time for a fresh approach was right, because this fantastic subject was only revealed to math majors--we needed to open linear algebra to the world. Those years of teaching led to the Haimo Prize from the Mathematical Association of America. For encouraging education worldwide, the International Congress of Industrial and applied Mathematics awarded me the first Su Buchin Prize. I am extremely grateful more than I could possibly say. What i hope most is that you will like linear algebra

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