[Math] Shilov’s Linear Algebra as a first linear algebra texbook

Question

I graduated from high school three years ago and I am plannig on going to university next year to study mathematics.
Since I have a year to prepare, I would like to learn some first year university subjects, like linear algebra. Is Shilov's "Linear Algebra", suitable for this purpose or is it too advanced ?

Answer 1

I haven't bought this book, but I have access to a copy of it. On a cursory read, I think it isn't too advanced and its choices of topics are about right. It covers slightly more topics than one shall see in a typical modern introductory course that focuses only on the theory of linear algebra.

Whether the book is suitable for beginners depends very much on the readers. There are books (like Rudin's Principles of Mathematical Analysis or Spivak's Calculus on Manifolds) that are praised by many people but also found too hard by equally many. To me, Shilov's exposition looks clear, efficient and rigourous. It is harder (but not to a large extent) than most modern introductory texts on linear algebra, but much easier to read than Hoffman and Kunze's classic, Linear Algebra.

Shilov introduces and discusses determinants in chapter 1. This is usually one of the most difficult topics in an introductory course. If you can get past chapter 1 without difficulty, you should be able to understand the whole book (sans the chapters marked by asterisks).

However, given that the book was published in the 1970s, those readers who are accustomed to the new breed of more "conversational" texts may dislike its style. They may also find it lacking motivations or practical applications.

If you are currently interested only in the theory (but not applications) of linear algebra, I think Berberian's Linear Algebra is another inexpensive and viable choice for private study. Jim Hefferon's Linear Algebra also worths a look. It does cover some applications and it is a favourite text of Darij Grinberg, a well respected user of this site. The electronic copy is free and the paper copy is cheap.