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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.Basically, there are two parts in 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. Tensor product of modules are also briefly discussed.
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.Basically, there are two parts in 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. Tensor product of modules are also briefly discussed.
This textbook provides an introduction to abstract algebra for
advanced undergraduate students. Based on the authors' notes at the
Department of Mathematics, National Chung Cheng University, it
contains material sufficient for three semesters of study. It
begins with a description of the algebraic structures of the ring
of integers and the field of rational numbers. Abstract groups are
then introduced. Technical results such as Lagrange's theorem and
Sylow's theorems follow as applications of group theory. The theory
of rings and ideals forms the second part of this textbook, with
the ring of integers, the polynomial rings and matrix rings as
basic examples. Emphasis will be on factorization in a factorial
domain. The final part of the book focuses on field extensions and
Galois theory to illustrate the correspondence between Galois
groups and splitting fields of separable polynomials.Three whole
new chapters are added to this second edition. Group action is
introduced to give a more in-depth discussion on Sylow's theorems.
We also provide a formula in solving combinatorial problems as an
application. We devote two chapters to module theory, which is a
natural generalization of the theory of the vector spaces. Readers
will see the similarity and subtle differences between the two. In
particular, determinant is formally defined and its properties
rigorously proved.The textbook is more accessible and less
ambitious than most existing books covering the same subject.
Readers will also find the pedagogical material very useful in
enhancing the teaching and learning of abstract algebra.
A First Course in Linear Algebra is written by two experts from
algebra who have more than 20 years of experience in algebra,
linear algebra and number theory. It prepares students with no
background in Linear Algebra. Students, after mastering the
materials in this textbook, can already understand any Linear
Algebra used in more advanced books and research papers in
Mathematics or in other scientific disciplines.This book provides a
solid foundation for the theory dealing with finite dimensional
vector spaces. It explains in details the relation between linear
transformations and matrices. One may thus use different viewpoints
to manipulate a matrix instead of a one-sided approach. Although
most of the examples are for real and complex matrices, a vector
space over a general field is briefly discussed. Several optional
sections are devoted to applications to demonstrate the power of
Linear Algebra.
A First Course in Linear Algebra is written by two experts from
algebra who have more than 20 years of experience in algebra,
linear algebra and number theory. It prepares students with no
background in Linear Algebra. Students, after mastering the
materials in this textbook, can already understand any Linear
Algebra used in more advanced books and research papers in
Mathematics or in other scientific disciplines.This book provides a
solid foundation for the theory dealing with finite dimensional
vector spaces. It explains in details the relation between linear
transformations and matrices. One may thus use different viewpoints
to manipulate a matrix instead of a one-sided approach. Although
most of the examples are for real and complex matrices, a vector
space over a general field is briefly discussed. Several optional
sections are devoted to applications to demonstrate the power of
Linear Algebra.
New Edition: A Course on Abstract Algebra (2nd Edition)This
textbook provides an introduction to abstract algebra for advanced
undergraduate students. Based on the authors' lecture notes at the
Department of Mathematics, National Chung Cheng University of
Taiwan, it begins with a description of the algebraic structures of
the ring and field of rational numbers. Abstract groups are then
introduced. Technical results such as Lagrange's Theorem and
Sylow's Theorems follow as applications of group theory. Ring
theory forms the second part of abstract algebra, with the ring of
polynomials and the matrix ring as basic examples. The general
theory of ideals as well as maximal ideals in the rings of
polynomials over the rational numbers are also discussed. The final
part of the book focuses on field theory, field extensions and then
Galois theory to illustrate the correspondence between the Galois
groups and field extensions.This textbook is more accessible and
less ambitious than most existing books covering the same subject.
Readers will also find the pedagogical material very useful in
enhancing the teaching and learning of abstract algebra.
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