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醉染图书线代数(第5版)9787302535560
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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
EliminationUsingMatrices 58
2 4 RulesforMatrixOraios 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
3 3 The Complete Solution to Ax = b 149
3 4 Independence,BasisandDimension 163
3 5 DimensionsoftheFourSubspaces 180
4 Orthogonality 193
4 1 OrthogonalityoftheFourSubspaces 193
4 2 Projections 205
4 3 LeastSquaresApproximations 218
4 4 OrthonormalBasesanGa-Schmidt 2
5 Determinants 246
5 1 ThePropertiesofDeterminants 246
5 2 PermutationsandCofactors 257
5 3 Cramer’sRule,Inverses,andVolumes 272
vii
6 Eigenvalues and Eigenvectors 287
6 1 IntroductiontoEigenvalues 287
6 2 DiagonalizingaMatrix 303
6 3 SystemsofDifferentialEquations 318
6 4 SymmetricMatrices 337
6 5 PositiveDe niteMatrices 349
7 TheSingularValueDecoition (SVD) 363
7 1 ImageProcessingbyLinearAlgebra 363
7 2 BasesandMatricesintheSVD 370
7 3 Principal Component Analysis (PCA by the SVD) 381
7 4 TheGeometryoftheSVD 391
8 LinearTransformations 400
8 1 TheIdeaofaLinearTransformation 400
8 2 TheMatrixofaLinearTransformation 410
8 3 TheSearchforaGoodBasis 420
9 ComplexVectorsand Matrices 429
9 1 ComplexNumbers 430
9 2 HermitianandUnitaryMatrices 437
9 3 TheFastFourierTransform 444
10 Applications 451
10 1GraphsandNetworks 451
10 2MatricesinEngineering 461
10 3 Markov Matrices, Population, and Economics 473
10 4LinearProgramming 482
10 5 Fourier Series: Linear Algebra for Functions 489
10 6ComputerGraphics 495
10 7LinearAlgebraforCryptography 501
11 NumericalLinear Algebra 507
11 1GaussianEliminationinPractice 507
11 2NormsandConditionNumbers 517
11 3 IterativeMethodsandPreconditioners 5
12LinearAlgebrain Probability& Statistics 534
12 1Mean,Variance,andProbability 534
12 2 Covariance Matrices and Joint Probabilities 545
1 Multivariate Gaussian and Weighted Least Squares 554
MatrixFactorizations 562
Index 564
SixGreatTheorems/LinearAlgebrain aNutshell 573
"作者GILBERT STRANG为Massachusetts Institute of Technology数学系教授。从UCLA博士后一直在MIT任教.教授的课程有“数据分析的矩阵方法”“线代数”“计算机科学与工程”等,出版的图书有Linear Algebra and Learning from Data (NEW)、See mat.i.eu/learningfromdata、Introduction to Linear Algebra - Fifth Edition 、Contact linearalgebrabook@gmail.com、Complete List of Books and Articles、Differential Equations and Linear Algebra。
"
Preface
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 mat.i.eu/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’hvet 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.
mat.i.eu/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 an Ja 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 oraio 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 besndmst 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 b an 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 tested on Ax = b: pure linear algebra. But even a supercomputer doesn’t want the inverse matrix: too slow. Inverses give the simplest formula x = A.1b 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 .rst 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. This book moves gradually and steadily from numbers to vectors to subspaces—each level comes naturally and everyone can get it.
Here are 12points about learning and teaching from this book :
1.
Chapter 1 starts with vectors and dot products. If the class has met them before, focus quickly on linear combinations. Section 1.3 provides three independent vectors whose combinations .ll all of 3-dimensional space, and three dependent vectors in a plane. Those two examples are thebeginning oflinear algebra.
2.
Chapter 2 shows the row picture and the column picture of Ax = b. The heart of linear algebra is in that connection between the rows of A and the columns of A: 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 is to capture the whole process—start with A, multiply by E’s, end with U.
Elimination is seen in the beautiful form A = LU. The lower triangular L holds the forward elimination steps, and U is upper triangular 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.
With more equations than unknowns, it is almost sure that Ax = b has no solution. We cannot throw out every measurement that is close but not perfectly exact! When we solve by least squares, the key will be the matrix ATA. This wonderful matrix appears everywhere in applied mathematics, when Ais rectangular.
5.
Determinants give formulas for all thahscme before—Cramer’s Rule, inverse matrices, volumes in n dimensions. We don’t need those formulas to com-pute. They slow us down. But detA =0 tells when a matrix is singular : this is the key to eigenvalues.
6.
Section 6.1 explains eigenvalues for 2 by 2 matrices. Many courses want to see eigenvalues early. It is completely reasonable to come here directly from Chapter 3, because the determinant is easy for a 2by 2 matrix. Thekey equationis Ax = λx.
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 λ) and the problem is one-dimensional.
An essential highlight of Chapter 6 is diagonalizing a symmetric matrix. When all the eigenvalues are positive, the matrix is “positive de.nite”. This key idea connects the whole course—positive pivots and determinants and eigenvalues and energy. I work hard to reach this point in the book and to explain it by examples.
7.
Chapter 7 is new. It introduces singular values and singular vectors. They separate all matrices into simple pieces, ranked in order of their importance. You will see one way to compress an image. Especially you can analyze a matrix full of data.
8.
Chapter 8 explains linear transformations. This is geometry without axes, algebra with no coordinates. When we choose a basis, we reach the best possible matrix.
9.
Chapter 9 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.1) is revolutionary.
10.
Chapter 10is full of applications, more than any single course could need:
10.1 Graphs andNetworks—leading to the edge-node matrix for Kirchhoff’s Laws
10.2 MatricesinEngineering—differential equations parallel to matrix equations
10.3 MarkovMatrices—as in Google’s PageRank algorithm
10.4 LinearProgramming—a new requirement x ≥ 0and minimization of the cost
10.5 FourierSeries—linear algebra for functions and digital signal processing
10.6 ComputerGraphics—matrices move and rotate and compress images
10.7 Linear Algebra in Cryptography—this new section was fun to write. The Hill Cipher is not too secure. It uses modular arithmetic: integers from 0 to p. 1. Multiplication gives 4 × 5 ≡ 1(md 9). For decoding this gives 4.1 ≡ 5.
11.
How should computing be included in a linear algebra course? Icnpen a new understanding of matrices—every class will .nd a balance. MATLAB and Mapleand Mathematica are powerful in different ways. Julia and Python are free and directly accessible on the Web. Those newer languages are powerful too !
Basic commands begin in Chapter 2. Then Chapter 11moves toward professional al-gorithms.You can upload and download codes for this course on the website.
12.
Chapter 12 on Probability and Statistics is new, with truly important applications. When random variables are not independent we get covariance matrices. Fortunately they are symmetric positive de.nite. The linear algebra in Chapter 6 is needed now.
Symmetric matrix Orthogonalmatrix Triangular matrix
. 2 .1 0 0. .1 1 1 1. .1 1 1 1.
... .1 0 2 .1 .1 2 0 .1 ... 1 2 ... 1 1 .1 1 1 .1 .1 .1 ... ... 0 0 1 0 1 1 1 1 ...
0 0 .1 2 1 .1 .1 1 0 0 0 1
The Variety of Linear Algebra
Calculus is mostly about one special oraio (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 .nd a light-hearted essay called “Too Much Calculus” on my website.
The truthis that vectors and matriceshavebecome thelanguage toknow.
Part of that language is the wonderful variety of matrices. Let me give three examples:
A key goal is learning to “read” a matrix. You need to see the meaning in the numbers. This is really the essence of mathematics—patterns and their meaning.
I have used italics and boldface to pick out the key words on each page. I know there are times when you want to read quickly, looking for the important lines.
May I end with this thought for professors. You might feel that the direction is righ,ndwnder 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
The greatest encouragement of all is the eein 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. Thankyou all.
One person has helped with every word in this book. He is Ashley C. Fernandes, who prepared the LATEX .les. It is now six books that he has allowed me to write and rewrite, aiming for accuracy and also for life. Working with friends is a happy way to live.
Friends inside and outside the MIT math department have been wonderful. Alan Edelman for Julia and much more, Alex Townsend for the .ag examples in 7.1, and Peter Kempthorne for the .nance example in 7.3: those stand out. Don Spickler’s website on cryptography is simply excellent. I thank Jon Bloom, Jack Dongarra, Hilary Finucane, Pavel Grinfeld, Randy LeVeque, David Vogan, Liang Wang, and Karen Willcox. The “ienaces” in 7.3 came from Matthew Turk and Jeff Jauregui. And the big step to singular values was accelerated by Raj Rao’s great course at Michigan.
This book owes so much to my happy sabbatical in Oxford. Thank you, Nick Trefethen and everyone. Especially you the reader! Best wishes in your work.
Background of the Author
This is my 9th textbook on linear algebra, and I hesitate to write about myself. It is the mathematics that is important, and the reader. The next paragraphs add something brief and 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 6 million when you include the videos on ocw.mit.edu). The time for a fresh approach was right, because this fantastic subjecwsnly revealed to math majors—we needed to open linear algebra to the world.
I am so grateful for a life of teaching mathematics, more than I could possibly tell you.
Gilbert Strang
Linear Algebra and Learning from Data This is the new textbook for the applied
linear algebra course 18.065at MIT. It starts with the basic factorizations of a matrix :
T
A= LU A= R A= XΛX.1 S= ΛT A= UΣVA= CMR
The last three are fundamental for data analysis—eigenvalues Λof a symmetric matrix
S, singular values Σ of any matrix A (the SVD is a highlight of this book too), and
column-mixing-row factors A = CMR.
Thegoalofdeeplearningisto .ndpatternsinthetrainingdata. Matrix multiplication
is interwoven with the nonlinear ramp function ReLU (x)= max (0,x). The result is
a learning function that can interpret new data. The textbook explains how and why
this succeeds—even in the classroom. Linear algebra and student projects are the keys.
t
THE MATRIX ALPHABET
A Any Matrix P Permutation Matrix B Basis Matrix P Projection Matrix C Cofactor Matrix Orthogonal Matrix D Diagonal Matrix R Upper Triangular Matrix E Elimination Matrix R Reduced Echelon Matrix F Fourier Matrix S Symmetric Matrix H Hadamard Matrix T Linear Transformation I Identity Matrix U Upper Triangular Matrix J Jordan Matrix U Left Singular Vectors K Stiffness Matrix V Right Singular Vectors L Lower Triangular Matrix X Eigenvector Matrix M Markov Matrix Λ Eigenvalue Matrix N Nullspace Matrix Σ Singular Value Matrix
Gilbert Strang的《线代数(第5版)》是一本经典线代数教材。此书深入浅出地展示了线代数的所有核心概念,讲述过程中恰当穿插了各种应用,体现了线代数特别有用的思想。
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