Advanced Linear Algebra: With an Introduction to Module Theory (Paperback)

Certain essential concepts in linear algebra cannot be fully explained in a first course. This is due to a lack of algebraic background for most beginning students. On the other hand, these concepts are taken for granted in most of the mathematical courses at graduate school level. This book will provide a gentle guidance for motivated students to fill the gap. It is not easy to find other books fulfilling this purpose. This book is a suitable textbook for a higher undergraduate course, as well as for a graduate student's self-study. The introduction of set theory and modules would be of particular interest to students who aspire to becoming algebraists.

There are three parts to this book. One is to complete the discussion of bases and dimension in linear algebra. In a first course, only the finite dimensional vector spaces are treated, and in most textbooks, it will assume the scalar field is the real number field. In this book, the general case of arbitrary dimension and arbitrary scalar fields is examined. To do so, an introduction to cardinality and Zorn's lemma in set theory is presented in detail. The second part is to complete the proof of canonical forms for linear endomorphisms and matrices. For this, a generalization of vector spaces, and the most fundamental results regarding modules are introduced to readers. This will provide the natural entrance into a full understanding of matrices. Finally, tensor products of vector spaces and modules are briefly discussed.

線性代數中的某些基本概念在初級課程中無法完全解釋清楚。這是因為大多數初學者缺乏代數背景。另一方面，這些概念在大多數研究生數學課程中被視為理所當然。本書將為有動機的學生提供溫和的指導，以填補這一空白。很難找到其他能夠實現這一目的的書籍。本書既適合高年級本科課程的教材，也適合研究生自學。對於有志於成為代數學家的學生來說，集合論和模的介紹尤其有趣。

本書分為三個部分。第一部分是完成線性代數中基底和維度的討論。在初級課程中，只處理有限維向量空間，並且在大多數教科書中，假設標量域是實數域。在本書中，將檢查任意維度和任意標量域的一般情況。為此，詳細介紹了集合論中的基數和佐恩引理。第二部分是完成線性自同態和矩陣的規範形式的證明。為此，向讀者介紹了向量空間的一般化以及關於模的最基本結果。這將為對矩陣的全面理解提供自然入口。最後，簡要討論了向量空間和模的張量積。