Linear Algebra: Theorems and Applications

Welcome to the beginning of this book - an algebra book that shows and explains further topics than those that are covered in a primary course on linear algebra. Such course usually leaves the student with the question “Why in the world do I have to go through linear algebra?” which is absolutely understandable. Algebra, and other sciences in general, are not entities that one just needs to pass as a course. Sciences are masters that stay with you and are always there whenever you need them, that in some occasions we do not understand that those masters are something else.

Before talking about the importance of algebra, let us first clear some details about this text. This book does not only contain a set of complicated equations with heavy definitions about abstract entities and problems that arise in several fields of mathematics, which in many other texts, makes you feel like you’re reading Mandarin. This book also contains clear and deductive explanations from different perspectives about those set of complicated equations, with the hope of making them accessible and easy to assimilate for the reader.

Algebra is the science of representing life through equations while keeping it simple. The reason for writing this book was, at first, the decision to spread knowledge of such a great science that is usually feared by many young students and ignored by others, when in reality Algebra is a body of knowledge so beautiful within itself, as well as necessary dealing with or approaching many mathematical situations. As this book is being developed, another reason to continue to write it arose – there is a lack of easy-to-understand texts, which are supposed to be the high end books that today’s generation are obligated to read, in order to understand the subjects they should know to continue their formation in their respective careers.

This book is dedicated to those who really want to learn linear algebra. But be warned, the journey is not promised to be easy as it will require dedication, honesty and time from the reader; but the journey will also be nurturing and revealing in the sense that at the end of each chapter, the reader will be able to say “I’ve got it!” in much the same way when Archimedes said “Eureka!” which means “I’ve found it!” In addition, this book also contains techniques and algorithms, along with their derivations and examples, which are used in the real world, thus helping the reader become a well prepared professional.

Many texts, articles and lessons were used as resources for this book, however, the most used resource in the development of this text was common sense. You see, mathematics is a free science, a body of knowledge that anyone can derive on its own. This book is just a compendium of the works of many mathematicians through the centuries, making it possible for humankind to get to where it is today. And as life goes on, it is probable that further developments, researches and advances will continue. That was one of the main ideas in the background while writing this text - to make the reader able to fully understand as much knowledge in linear algebra as possible, especially those concerning methods and algorithms that arose in the twentieth century.

Just as a brief comment, the first chapter of this book is dedicated to making certain topics clear to avoid confusion, and answering questions that usually arise during the first course in linear algebra. The rest of the chapters are more advanced and developed.

Claire Montemar is a licensed Chemical Engineer with a real passion for teaching, learning and taking care of the environment. She obtained her degree from the University of the Philippines in 1999. She has worked in the manufacturing industry for a few years but spent most of her career as an academic tutor/instructor for elementary, high school and college students. Her forte is in the areas of mathematics and science, particularly algebra and chemistry. She currently works as a full-time freelancer, predominantly serving the needs of clients in the education industry.