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.
Contents:

[*]Modules and Vector Spaces
[*]Linear Maps
[*]Determinant
[*]Canonical Forms
[*]A Brief Introduction to the Tensor Product

Readership: Advanced undergraduates (junior or senior) or for first-year graduate students majoring in pure maths. This book is ideal for those taking a continuation course following a first course on linear algebra and a first course on abstract algebra.
Key [b]Features:[/b]

[*]One can hardly find other similar books in the market. This book collects materials from many areas of mathematics and presents them in a concise manner to help students understand the topics of the book in a most efficient way
[*]This book was originally a series of lecture notes and had been tested on targeted readers
[*]This book contains well-chosen subjects and well-designed sets of exercises
[*]This book will help answer most questions which naturally arise when a student completes a first course in linear algebra