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 includes an emphasis on 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 for a reader who is (1) planning to pursue mathematics further, 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 consisted of students with experience in mathematical competitions who wanted to learn college-level mathematics. There are minimal prerequisites for reading this book, aside from fluency in mathematical communication, i.e., reading and writing proofs.

I tried to minimize the material covered 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 is arguably 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.

Organization

In Chapter 1, we introduce naïve 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 sums, 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 to 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 exercises 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 practice 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 a set of linear algebra notes remained. I would like to thank everyone who 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 rolling draft. I welcome reports of typos and mathematical errors, as well as suggestions, comments, and criticisms, through the issue tracker. The latest web edition is available on the project website, and the source files are available on GitHub. Happy reading!