A rough guide to linear algbera Dongryul Kim Version of April 18, 2020 ii Contents Preface v 1 Sets 1 1.1 Sets and maps . .1 1.2 Products, coproducts, and sets of maps . .4 1.3 Fun with diagrams . .7 1.4 Equivalence classes . .9 2 Vector spaces 13 2.1 Fields . 13 2.2 Vector spaces . 15 2.3 Matrices . 18 2.4 Products and direct sums . 21 2.5 Subspaces and quotients . 27 2.6 Vector spaces from linear maps . 29 2.7 Bases and dimension . 34 2.8 Dual spaces . 44 2.9 Linear algebra in combinatorics . 50 3 Multilinear algebra 55 3.1 Bilinear maps and tensor products . 55 3.2 Symmetric and exterior algebras . 62 3.3 The determinant . 68 3.4 Computing the inverse matrix . 72 4 Linear algebra without division 79 4.1 Commutative rings . 79 4.2 Modules . 82 4.3 Classification of finitely generated modules over a PID . 86 4.4 Frobenius and Jordan normal form . 97 4.5 Eigenvalues and eigenvectors . 103 iii iv CONTENTS 5 Linear algebra over R and C 109 5.1 A bit of analysis . 109 5.2 Inner products . 112 5.3 Operators on Hilbert spaces . 118 5.4 The spectral theorem . 121 5.5 Positivity of operators . 125 5.6 Duality in linear programming . 130 Epilogue 131 Preface At a certain point, one adopts a mode of learning mathematics. Then at a later point, after progressing through various styles of teaching and writing, one develops one's own point of view for how mathematics is to be taught. This book is an arrogant attempt to reassemble linear algebra according to the author's pedagogy. About this book There are countless (seriously, too many) books on linear algebra, and it is necessary to ask why we need another. Most of these books are designed for a first course in linear algebra for college students, which possibly include an emphasis in proof writing. However, there is a definite discrepancy between the presentation of linear algebra in such textbooks and how mathematicians understand the theory. Almost every professional mathematician has a natural functorial picture of linear algebra, but this is rarely written down in textbooks. Hence in reality, one acquires the functorial understanding while learning more advanced mathematics, such as commutative algebra or vector bundles. The philosophy of this book is that the determined reader can benefit from being directly introduced to the abstract development of linear algebra. Hence this book will be most suitable to a reader who is (1) planning to pursue a career in mathematics, and (2) is already comfortable with rigorous presentations of mathematics. I believe it can also be used for a second course in linear algebra. The audience I had in mind when writing the book are students who had experience in mathematical competitions wanting to learn college-level mathematics. There are minimal prerequisites for reading this book, aside from fluency in mathematical communications, i.e., reading and writing proofs. I tried to minimize the material dealt in this book. My conception of the book is that it is the bare bones of linear algebra. Most mathematicians, working in all fields, will be familiar with all the material in this book. Linear algebra arguably is the main tool for studying mathematical objects. On the other hand, the book will serve as a solid background for learning other parts of mathematics. I have tried to explain some of the applications of linear algebra in the epilogue. v vi CONTENTS Organization In Chapter 1, we introduce na¨ıve set theory. Set theory provides the foundation of modern mathematics, and it is necessary that the reader is familiar with sets and the basic operations. We have also included a first taste of category theory, introducing the concept of commutative diagrams and universal properties. In Chapter 2, we introduce the theory of vector spaces. Linear algebra is, and should be, the study of vector spaces and linear maps, not of matrices. Matrices are a good tool for computations, but abstraction is necessary for a solid conceptual picture. During the first half of the chapter, we carefully develop various constructions in the category of vector spaces, e.g., products, direct sum, subspaces, quotient spaces, kernel, cokernel, image, internal hom, etc. In the second half, we prove the theorem that every vector space has a basis, and discuss applications of this fact. We will not ignore infinite-dimensional spaces. There is also a section on applications of linear algebra in combinatorics, for students with a competition mathematics background. In Chapter 3, we introduce tensor products, the symmetric algebra, and the exterior algebra. We discuss the universal properties of these vector spaces as a motivation. The determinant is defined using the exterior algebra, as it should be. Using this definition, we prove Cramer's rule and discuss Gaussian elimination. In Chapter 4, we introduce modules over a commutative ring. Modules are important objects in commutative algebra, but our main goal in this chapter is study the different possible ways a linear map can act on a vector space. We prove the classification theorem of finitely generated modules over a principal ideal domain, and use this to discuss Frobenius normal form, Jordan normal form, eigenvectors, eigenspaces, and generalized eigenspaces. In Chapter 5, we do linear algebra combined with analysis. We define inner product structures and finite-dimensional Hilbert spaces, and lead up to proving the spectral theorem on finite-dimensional spaces. There is a more algebraic proof in the finite-dimensional case, using the Jordan normal form for instance, but we use an analytic argument that applies also to compact operators on infinite-dimensional Hilbert spaces. There are plenty of exercises. I have tried to meld the exercise with the text so that they appear in context. Some of the exercises will be used in later proofs, and some of them will be used implicitly throughout the book. Others will be unimportant from a theoretical perspective, but nonetheless helpful in properly understanding the material. Abstraction can be a double-edged sword, as the abstract loses meaning when disconnected from the concrete. By doing the exercises, the reader is expected to train moving between the two worlds. Acknowledgements This project started with a small series of lectures delivered to the Korean team for the 2017 International Mathematical Olympiad, a group of six high school students. The course was aborted due to lack of time, but the idea of writing CONTENTS vii a set of linear algebra notes remained. I would like to thank everyone who had attended my ill-prepared lectures, as well as those who were beside me when I was working on the book. The current text is a draft, and will be updated occasionally. I beg ev- eryone to send me typos, grammatical errors, mathematical errors, sugges- tions, comments, criticisms, anything about the book. You can reach me at [email protected]. The latest version of this book can be obtained at my website http://dongryulkim.wordpress.com. Happy reading! April, 2020 Dongryul Kim Here, I keep a list of chores I need to do. The draft will be a final draft once the list is empty. • Probably expand the chapter \Preface" once I indicate optional/required material • Do we move where we discuss Gaussian elimination? Maybe matrices and algorithms deserve a separate section? • Write about the LU decomposition • Include somewhere a discussion Cayley{Hamilton • Fill in the section \Duality in linear programming" • Introduce functional calculus for self-adjoint operators • Check which are optional, which are required • Rank exercises by difficulty? • Give hints for exercises? • Add references at the end of each chapter? viii CONTENTS Chapter 1 Sets Most of mathematics is grounded on the notion of sets. In this chapter, we quickly review the basic set theory we use throughout the book, assuming that the reader is familiar with most of the material. 1.1 Sets and maps In standard mathematics, that is, Zermelo{Fraenkel set theory, literally every mathematical object is a set. Each of the numbers 0; 1; 2;::: is actually a set: 0 = ; = fg; 1 = f;g = ffgg; 2 = f;; f;gg = ffg; ffggg;:::: But because the notion of sets provides a foundation for mathematics, it does not make sense to mathematically define what a set is. Instead, people take the notion as granted and assume that sets satisfy certain axioms. But we are not going to worry to much about these issues and the following definition will be good for us. Definition 1.1.1. A set is a well-defined collection of objects. If an object x is inside the set X, we say that x is an element of X and write x 2 X. Example 1.1.2. The set of natural numbers Z≥0 = f0; 1; 2;:::g is a set. The set of integers Z = f:::; −1; 0; 1;:::g is a set. The set of even integers 2Z = f:::; −2; 0; 2; 4;:::g = f2x : x 2 Zg is a set. An issue arises when one tries to look at the set of all sets. In fact, in Zermelo{Fraenkel set theory, it is possible to prove that there is no set that contains all sets. But again, we are going to ignore such issues. Definition 1.1.3. Given two sets X and Y , if x 2 Y implies x 2 X, we say that Y is a subset of X and write Y ⊆ X. Equivalently, we say that X is a superset of Y and write X ⊇ Y . For instance, f1; 3g ⊆ f1; 2; 3g.