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# Linear Algebra Done Right

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the Instructor Major changes from the previous edition: � This edition contains 561 exercises, including 337 new exercises that were not in the previous edition. Exercises now appear at the end of each section, rather than at the end of each chapter. � Many new examples have been added to illustrate the key ideas of linear algebra. � Beautiful new formatting, including the use of color, creates pages with an unusually pleasant appearance in both print and electronic versions. As a visual aid, deﬁnitions are in beige boxes and theorems are in blue boxes (in color versions of the book). � Each theorem now has a descriptive name. � New topics covered in the book include product spaces, quotient spaces, and duality. � Chapter 9 (Operators on Real Vector Spaces) has been completely rewritten to take advantage of simpliﬁcations via complexiﬁcation. This approach allows for more streamlined presentations in Chapters 5 and 7 because those chapters now focus mostly on complex vector spaces. � Hundreds of improvements have been made throughout the book. For example, the proof of Jordan Form (Section 8.D) has been simpliﬁed. Please check the website below for additional information about the book. I may occasionally write new sections on additional topics. These new sections will be posted on the website. Your suggestions, comments, and corrections are most welcome. Best wishes for teaching a successful linear algebra class! Sheldon Axler Mathematics Department San Francisco State University San Francisco, CA 94132, USA website: linear.axler.net e-mail: linear@axler.net Twitter: @AxlerLinear Preface for the Student You are probably about to begin your second exposure to linear algebra. Unlike your ﬁrst brush with the subject, which probably emphasized Euclidean spaces and matrices, this encounter will focus on abstract vector spaces and linear maps. These terms will be deﬁned later, so don’t worry if you do not know what they mean. This book starts from the beginning of the subject, assuming no knowledge of linear algebra. The key point is that you are about to immerse yourself in serious mathematics, with an emphasis on attaining a deep understanding of the deﬁnitions, theorems, and proofs. You cannot read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast. When you encounter the phrase “as you should verify”, you should indeed do the veriﬁcation, which will usually require some writing on your part. When steps are left out, you need to supply the missing pieces. You should ponder and internalize each deﬁnition. For each theorem, you should seek examples to show why each hypothesis is necessary. Discussions with other students should help. As a visual aid, deﬁnitions are in beige boxes and theorems are in blue boxes (in color versions of the book). Each theorem has a descriptive name. Please check the website below for additional information about the book. I may occasionally write new sections on additional topics. These new sections will be posted on the website. Your suggestions, comments, and corrections are most welcome. Best wishes for success and enjoyment in learning linear algebra! Sheldon Axler Mathematics Department San Francisco State University San Francisco, CA 94132, USA website: linear.axler.net e-mail: linear@axler.net Twitter: @AxlerLinear xv Acknowledgments I owe a huge intellectual debt to the many mathematicians who created linear algebra over the past two centuries. The results in this book belong to the common heritage of mathematics. A special case of a theorem may ﬁrst have been proved in the nineteenth century, then slowly sharpened and improved by many mathematicians. Bestowing proper credit on all the contributors would be a difﬁcult task that I have not undertaken. In no case should the reader assume that any theorem presented here represents my original contribution. However, in writing this book I tried to think about the best way to present lin- ear algebra and to prove its theorems, without regard to the standard methods and proofs used in most textbooks. Many people helped make this a better book. The two previous editions of this book were used as a textbook at about 300 universities and colleges. I received thousands of suggestions and comments from faculty and students who used the second edition. I looked carefully at all those suggestions as I was working on this edition. At ﬁrst, I tried keeping track of whose suggestions I used so that those people could be thanked here. But as changes were made and then replaced with better suggestions, and as the list grew longer, keeping track of the sources of each suggestion became too complicated. And lists are boring to read anyway. Thus in lieu of a long list of people who contributed good ideas, I will just say how truly grateful I am to everyone who sent me suggestions and comments. Many many thanks! Special thanks to Ken Ribet and his giant (220 students) linear algebra class at Berkeley that class-tested a preliminary version of this third edition and that sent me more suggestions and corrections than any other group. Finally, I thank Springer for providing me with help when I needed it and for allowing me the freedom to make the ﬁnal decisions about the content and appearance of this book. Special thanks to Elizabeth Loew for her wonderful work as editor and David Kramer for unusually skillful copyediting. Sheldon Axler xvii CHAPTER 1 René Descartes explaining his work to Queen Christina of Sweden. Vector spaces are a generalization of the description of a plane using two coordinates, as published by Descartes in 1637. Vector Spaces Linear algebra is the study of linear maps on ﬁnite-dimensional vector spaces. Eventually we will learn what all these terms mean. In this chapter we will deﬁne vector spaces and discuss their elementary properties. In linear algebra, better theorems and more insight emerge if complex numbers are investigated along with real numbers. Thus we will begin by introducing the complex numbers and their basic properties. We will generalize the examples of a plane and ordinary space to Rn and Cn, which we then will generalize to the notion of a vector space. The elementary properties of a vector space will already seem familiar to you. Then our next topic will be subspaces, which play a role for vector spaces analogous to the role played by subsets for sets. Finally, we will look at sums of subspaces (analogous to unions of subsets) and direct sums of subspaces (analogous to unions of disjoint sets). LEARNING OBJECTIVES FOR THIS CHAPTER basic properties of the complex numbers Rn and Cn vector spaces subspaces sums and direct sums of subspaces © Springer International Publishing 2015 S. Axler, Linear Algebra Done Right, Undergraduate Texts in Mathematics, DOI 10.1007/978-3-319-11080-6__1 1 2 CHAPTER 1 Vector Spaces 1.A Rn and Cn Complex Numbers You should already be familiar with basic properties of the set R of real numbers. Complex numbers were invented so that we can take square roots of negative numbers. The idea is to assume we have a square root of �1, denoted i , that obeys the usual rules of arithmetic. Here are the formal deﬁnitions: 1.1 Deﬁnition complex numbers � A complex number is an ordered pair .a; b/, where a; b 2 R, but we will write this as aC bi . � The set of all complex numbers is denoted by C: C D faC bi W a; b 2 Rg: � Addition and multiplication on C are deﬁned by .aC bi/C .c C di/ D .aC c/C .b C d/i; .aC bi/.c C di/ D .ac � bd/C .ad C bc/i I here a; b; c; d 2 R. If a 2 R, we identify aC 0i with the real number a. Thus we can think of R as a subset of C. We also usually write 0C bi as just bi , and we usually write 0C 1i as just i . The symbol i was