作者简介 "作者GILBERT STRANG为Massachusetts Institute of Technology数学系教授。从UCLA博士毕业后一直在MIT任教.教授的课程有“数据分析的矩阵方法”“线性代数”“计算机科学与工程”等,出版的图书有Linear Algebra and Learning from Data (NEW)、See math.mit.edu/learningfromdata、Introduction to Linear Algebra - Fifth Edition 、Contact linearalgebrabook@gmail.com、Complete List of Books and Articles、Differential Equations and Linear Algebra。 "
目录 Table of Contents 1 Introduction to Vectors 1 1 1 VectorsandLinearCombinations 2
1 2 LengthsandDotProducts 11
1 3 Matrices 22
2 Solving Linear Equations 31 2 1 VectorsandLinearEquations 31
2 2 TheIdeaofElimination 46
2 3 EliminationUsingMatrices 58
2 4 RulesforMatrixOperations 70
2 5 InverseMatrices 83
2 6 Elimination = Factorization: A = LU 97
2 7 TransposesandPermutations 108
3 Vector Spaces and Subspaces 122 3 1 SpacesofVectors 122
3 2 The Nullspace of A: Solving Ax = 0and Rx =0 134
I am happy for you to see this Fifth Edition of Introduction to Linear Algebra. This is the text for my video lectures on MIT’s OpenCourseWare (ocw.mit.edu and also YouTube). I hope those lectures will be useful to you (maybe even enjoyable !). Hundreds of colleges and universities have chosen this textbook for their basic linear algebra course. A sabbatical gave me a chance to prepare two new chapters about probability and statistics and understanding data. Thousands of other improvements too— probably only noticed by the author. . . Here is a new addition for students and all readers: Every section opens with a brief summary to explain its contents. When you read a new section, and when you revisit a section to review and organize it in your mind, those lines are a quick guide and an aid to memory. Another big change comes on this book’s website math.mit.edu/linearalgebra. That site now contains solutions to the Problem Sets in the book. With unlimited space, this is much more .exible than printing short solutions. There are three key websites : ocw.mit.edu Messages come from thousands of students and faculty about linear algebra on this OpenCourseWare site. The 18.06 and 18.06 SC courses include 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 2 a.m. (The reader doesn’t have to be in a class at all.) Six million viewers around the world have seen these videos (amazing). I hope you .nd them helpful. web.mit.edu/18.06 This site has homeworks and exams (with solutions) for the current course as it is taught, and as far back as 1996. There are also review questions, Java demos, Teaching Codes, and short essays (and the video lectures). My goal is to make this book as useful to you as possible, with all the course material we can provide. math.mit.edu/linearalgebra This has become an active website. It now has Solutions to Exercises—with space to explain ideas. There are also new exercises from many dif-ferent sources—practice problems, development of textbook examples, codes in MATLAB and Julia and Python, plus whole collections of exams (18.06 and others) for review. Please visit this linear algebra site. Send suggestions to linearalgebrabook@gmail.com i
The Fifth Edition The cover shows the Four Fundamental Subspaces—the row space and nullspace are on the left side, the column space and the nullspace of AT are on the right. It is not usual to put the central ideas of the subject on display like this! When you meet those four spaces in Chapter 3, you will understand why that picture is so central to linear algebra. Those were named the Four Fundamental Subspaces in my .rst book, and they start from a matrix A. Each row of A is a vector in n-dimensional space. When the matrix has m rows, each column is a vector in m-dimensional space. The crucial operation in linear algebra is to take linear combinations of column vectors. This is exactly the result of a matrix-vector multiplication. Ax is a combination of the columns of A. When we take all combinations Ax of the column vectors, we get the column space. If this space includes the vector b, we can solve the equation Ax = b. May I call special attention to Section 1.3, where these ideas come early—with two speci.c examples. You are not expected to catch every detail of vector spaces in one day! But you will see the .rst matrices in the book, and a picture of their column spaces. There is even an inverse matrix and its connection to calculus. You will be learning the language of linear algebra in the best and most ef.cient way: by using it. Every section of the basic course ends with a large collection of review problems. They 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 Challenge Problems go a step further, and sometimes deeper. Let me give four examples: Section 2.1: Which row exchanges of a Sudoku matrix produce another Sudoku matrix? Section 2.7: If P is a permutation matrix, why is some power Pk equal to I? Section 3.4: If Ax = band Cx = bhave the same solutions for every b, does Aequal C? Section 4.1: What conditions on the four vectors r, n, c, . 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 producesb. The solution vector x comes at three levels and all are important: 1. Directsolution to .nd x by forward elimination and back substitution.
2. Matrix solutionusing the inverse matrix: x = A.1b (if Ahas an inverse).
3. Particular solution(to Ay = b) plus nullspace solution (to Az =0).
That vector space solution x = y+ z is shown on the cover of the book. Direct elimination is the most frequently used algorithm in scienti.c computing. The matrix Abecomes triangular—then solutions come quickly. We also see bases for the four subspaces. But don’t spend forever on practicing elimination . . . good ideas are coming. The speed of every new supercomputer is teste
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