Introduction to Linear Algebra, 5th edition--Gilbert Strang.pdf-资料库

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INTRODUCTI N TO LINEAR ALGEBRA Fifth Edition GILBERT STRANG Massachusetts Institute of Technology WELLESLEY -CAMBRIDGE PRESS Box 812060 Wellesley MA 02482

Introduction Copyright ©2016 by Gilbert ISBN 978-0-9802327- to Linear Algebra, Strang 7-6 5th Edition reserved. All rights by any means, including Press. Wellesley authorized are arranged BTEX typesetting Printed -Cambridge translations in the United States of America No part of this book may be reproduced from prohibited photocopying, without Translation permission is strictly or stored written or transmitted in any language by the publisher. - by Ashley C. Fernandes (info@problemsolvingpathway.com) QA184.S78 2016 512'.5 93-14092 9876543 Other texts from Wellesley -Cambridge Press Computational Science and Engineering , Gilbert Strang ISBN 978-0-9614088-1- 7 Wavelets and Filter Banks, Gilbert Strang and Truong Nguyen ISBN 978-0-9614088- 7-9 Introduction to Applied Mathematics, Gilbert Strang ISBN 978-0-9614088-0-0 Calculus Third Edition (2017), Gilbert Strang ISBN 978-0-9802327-5-2 Algorithms for Global Positioning, Kai Borre & Gilbert Strang ISBN 978-0-9802327-3-8 Essays in Linear Algebra, Gilbert Strang ISBN 978-0-9802327-6-9 Differential Equations and Linear Algebra, Gilbert Strang ISBN 978-0-9802327-9-0 An Analysis of the Finite Element Method, 2008 edition, Gilbert Strang and George Fix ISBN 978-0-9802327-0- 7 Press -Cambridge Wellesley Box 812060 Wellesley www.wellesleycambridge.com linearalgebrabook@gmail.com math.mit.edu/�gs phone(781)431-8488 fax (617) 253-4358 MA 02482 USA The website The Solution for this book is math.mit.edu/linearalgebra Manual can be printed from that website. . Course material are available including syllabus and exams and also videotaped lectures on the book website and the teaching website: web.mit.edu/18.06 is included Linear Algebra This provides MATLAB® is a registered video lectures of the full linear algebra course of The Math Works, Inc. trademark in MIT's OpenCourseWare site ocw.mit.edu 18.06 and 18.06 SC. . a central idea of linear algebra. The front cover captures Ax = bis solvable One particular Add any vector The complete solution z from the (green) solution when bis in the (red) column space of A. y is in the (yellow) row space: Ay = b. nullspace of A: Az = 0. is x = y + z. Then Ax = Ay + Az = b. The cover design was the inspiration of Lois Sellers and Gail Corbett.

Preface I am happy for you to see this Fifth Edition This is the text for my video lectures also YouTube). I hope those lectures will be useful on MIT's OpenCourseWare of Introduction to Linear Algebra. (ocw.mit.edu and to you (maybe even enjoyable !). Hundreds of coll�ges and universities have chosen this textbook for their basic linear A sabbatical gave me a chance to prepare two new chapters course. algebra probability probably and statistics and understanding data. Thousands of other improvements only noticed by the author. . . Here is a new addition for students and all readers: about too opens with a brief summary Every section read a new section, it in your mind, those lines are a quick guide and an aid to memory. its contents. When you to review to explain a section and when you revisit and organize big change comes on this book's to the Problem Another now contains much more flexible than printing solutions Sets in the book. With unlimited There are three key websites short solutions. space, this is : website math.mit.edu/linearalgebra. That site come from thousands of students and faculty about linear site. The 18.06 and 18.06 SC courses include algebra video lectures of Messages ocw.mit.edu on this OpenCourseWare a complete subject be 2 a.m. (The reader world have seen these videos semester doesn't of classes. based on this textbook-the professor's Those lectures offer an independent review time stays free and the student's time can around the viewers of the whole have to be in a class at all.) Six million (amazing). I hope you find them helpful. as it is taught, web.mit.edu/18.06 course Teaching as useful Codes, and short essays to you as possible, This site has homeworks and exams (with solutions) for the current and as far back as 1996. There are also review questions, Java demos, (and the video lecture s). My goal is to make this book we can provide. with all the course material math.mit.edu/linearalgebr space to explain to Exercises-with ferent sources-practice development problems, and Julia and Python, plus whole collections a This has become an active of textbook of exams (18.06 ideas. examples, codes in MATLAB and others) for review. There are also new exercises from many dif website. It now has Solutions Please visit this linear algebra site. Send suggestions to linearalgebrabook@gm ail.com V

vi Preface The Fifth Edition The cover shows the Four Fundamental on the left side, the column space and the nulls to put the central in Chapter ideas of the subject 3, you will understand why that picture Those were named the Four Fundamental on display Subspaces-the row space and nullspace are pace of AT are on the right. like this! When you meet those four spaces It is not usual from a matrix A. Each row of A is a vector has m rows, each column is a vector linear of a matrix-vector multiplication. is to take linear algebra in m-dimensional of column vectors. combinations Ax is a combination of the columns of A. space. The crucial This is exactly operation the result in Subspaces in n-dimensional in my first book, and they start space. When the matrix is so central to linear algebra. 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 You are not expected examples. specific But you will see the first matrices There is even an inverse matrix language of linear algebra and its connection to calculus. way: by using it. in the best and most efficient spaces in one day! of their column spaces. You will be learning the to catch every detail of vector in the book, and a picture Every section of the basic course ends with a large collection of review problems. ask you to use the ideas in that section--the that space, look for computations Challenge Problems go a step further, by hand on a small matrix, the rank and inverse and determinant and sometimes and eigenvalues and they have been highly of A. Many problems praised. The deeper. Let me give four examples: dimension of the column space, They a basis for Section 2.1: Which row exchanges of a Sudoku matrix produce another Sudoku matrix? Section 2.7: If Pis a permutation matrix, why is some power pk equal to I? Section 3.4: If Ax= band Cx = b have the same solutions for every b, does A equal C? Section the row space, 4.1: What conditions the nullspace, on the four vectors r, n, c, £ allow them to be bases for the column space, and the left nullspace of a 2 by 2 matrix? The Start of the Course The equation Ax is a combination b. The solution produces Ax = b uses the language of linear combinations of the columns of A. The equation is asking right away. The vector for a combination that 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 matrix: x = A-1b (if A has 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.

Preface vii Direct elimination is the most frequently triangular-then used algorithm come quickly. in scientific We also see bases for the four computing. The A becomes matrix subspaces. solutions on practicing But don't spend forever 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 formula x = A-lb but not the top speed. even slower-there determinant of an n by n matrix. is no way a linear And everyone must know that determinants algebra course should Those formulas have a place, are begin with formulas for the but not first place. want the inverse matrix: too slow. Inverses give the simplest Structure of the Textbook in this preface, Already to explain of linear numbers this beautiful algebra to vectors Here are 12 points you can see the style of the book and its goal. That goal is serious, and usefulpart You will see how the applications of mathematics. reinforce the key ideas. This book moves gradually to subspaces-each level comes naturally from this book and everyone : and teaching about learning and steadily can get it. from 1.Chapter 1 starts with vectors and dot products. If the class has met them before, focus quickly vectors vectors on linear combinations. Section 1.3 provides three independent whose combinations in a plane. Those two examples space, are the beginning fill all of 3-dimensional and three dependent of linear algebra. 2.Chapter 2 shows the row picture and the column picture of Ax = b. The heart of is in that connection linear algebra the same numbers an elimination the whole process-start between pictures. A to produce matrix E multiplies but very different with A, multiply by E's, end with U. the rows of A and the columns of of matrices: the algebra Then begins A : a zero. The goal is to capture Elimination the forward elimination steps, is seen in the beautiful form A = LU. The lower triangular L holds and U is upper triangular for back substitution. algebra 3.Chapter all linear columns the key information 3 is linear combinations are needed? of the columns. The answer tells at the best level: question is: How many of those The crucial column space, us the dimension of the of Linear Theorem about A. We reach the Fundamental subspaces. The column space contains Algebra. and 4.With more equations than unknowns, it is almost We cannot throw out every measurement When we solve by least squares, matrix the key will be the matrix mathematics, that is close but not perfectly exact! AT A. This wonderful when A is rectangular. sure that Ax = b has no solution. everywhere in applied appears 5.Determinants give formulas for all that has come before-Cramer's Rule, volumes matrices, inverse pute. They slow us down. But det A = 0 tells when a matrix the key to eigenvalues. We don't need those formulas to com : this is is singular inn dimensions.

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Introduction to Linear Algebra, 5th edition--Gilbert Strang.pdf

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