Linear algebra is one of the important subjects which are treated in many areassuch as the natural sciences, computer sciences, engineering, and economy, etc. Up to date linear algebra is also one of the useful theories that supply the fundamental and systematic methods to erect those academic areas developed diversly and deeply in scholarly pursuits. According to the various different types of requirements in those areas, the contents of linear algebra have been greatly influenced. It is really difficult work that one can obtain a book containing the useful contents with these all requirements for undergraduate students. As a foundation of such a book, the first edition of this book is edited toward the basic theories required in each academic area, and this book is of the form like a lecture note consisting of mainly theoretical aspects and was mainly written for a one semester course in linear algebra at the junior, senior, and sophomore undergraduate level.
The one of the most important aims of the book is to induce themselves to find the methods analyzing concretely vector structures of the finite dimensional vector spaces over the given ground fields, and was centered on making them understood important properties and concepts appearing in vector spaces having more complicated structures through the use of the methods. For the purpose the book provide sufficient examples to explain the meanings inside given definitions, lemmas, propositions, and theorems and help out to solve the exercises given in each section of the book. One may omit the sections having advanced concepts in chapters 6, 7, and 8, whenever one teaches junior undergraduate students without obstructing the flaw of the aim of the book.
The contents of the book can be classified broadly into three parts. First, the book contains basic concepts with respect to the ground fields of vector spaces. In fact many other linear algebra books avoid the details of the fields, not even its definition and the related basic facts with appropriate examples. However it follow from the definite meaning of the field that one can see more easily the structures of the vector spaces, and it is very important matter how the vector space is defined on the ground fields which consists of so called scalars. For this reason the book introduce the basic concepts of fields in chapter 1 and matrices defined from the given ground fields together with the related problems. In chapter 2 we investigate the basic concepts and structures of a general vector space over a field and try to expect the explicit geometric structures of vector spaces over the field through the three dimensional real vector space over the real number system. Next, in chapter 4, we study methods to analyze vector structures indirectly by using the linear transformations with the corresponding matrices over the fields, which preserve the given operations on the vector spaces. In chapter 6, to see more explicit examples for the vector structures of finite dimensional vector spaces, the inner product spaces is introduced together with the related properties. From the facts it is shown that every n-dimensional vector space over a field has the same vector structures as the n-dimension real vector space over the real number system.
