# Elements of Abstract and Linear Algebra

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About *Elements of Abstract and Linear Algebra:*

Excerpts from book:

This book is a survey of abstract algebra with emphasis on linear algebra. It is intended for students in mathematics, computer science, and the physical sciences. The first three or four chapters can stand alone as a one semester course in abstract algebra. However they are structured to provide the background for the chapter on linear algebra. Chapter 2 is the most difficult part of the book because groups are written in additive and multiplicative notation, and the concept of coset is confusing at first. After Chapter 2 the book gets easier as you go along. Indeed, after the first four chapters, the linear algebra follows easily. Finishing the chapter on linear algebra gives a basic one year undergraduate course in abstract algebra. Chapter 6 continues the material to complete a first year graduate course. Classes with little background can do the first three chapters in the first semester, and chapters 4 and 5 in the second semester. More advanced classes can do four chapters the first semester and chapters 5 and 6 the second semester. As bare as the first four chapters are, you still have to truck right along to finish them in one semester.

The presentation is compact and tightly organized, but still somewhat informal. The proofs of many of the elementary theorems are omitted. These proofs are to be provided by the professor in class or assigned as homework exercises. There is a non-trivial theorem stated without proof in Chapter 4, namely the determinant of the product is the product of the determinants. For the proper flow of the course, this theorem should be assumed there without proof. The proof is contained in Chapter 6. The Jordan form should not be considered part of Chapter 5. It is stated there only as a reference for undergraduate courses. Finally, Chapter 6 is not written primarily for reference, but as an additional chapter for more advanced courses.

This text is written with the conviction that it is more effective to teach abstract and linear algebra as one coherent discipline rather than as two separate ones. Teaching abstract algebra and linear algebra as distinct courses results in a loss of synergy and a loss of momentum. Also with this text the professor does not extract the course from the text, but rather builds the course upon it. I am convinced it is easier to build a course from a base than to extract it from a big book. Because after you extract it, you still have to build it. The bare bones nature of this book adds to its flexibility, because you can build whatever course you want around it. Basic algebra is a subject of incredible elegance and utility, but it requires a lot of organization. This book is my attempt at that organization. Every effort has been extended to make the subject move rapidly and to make the flow from one topic to the next as seamless as possible. The student has limited time during the semester for serious study, and this time should be allocated with care. The professor picks which topics to assign for serious study and which ones to "wave arms at". The goal is to stay focused and go forward, because mathematics is learned in hindsight.

This book is a survey of abstract algebra with emphasis on linear algebra. It is intended for students in mathematics, computer science, and the physical sciences. The first three or four chapters can stand alone as a one semester course in abstract algebra. However they are structured to provide the background for the chapter on linear algebra. Chapter 2 is the most difficult part of the book because groups are written in additive and multiplicative notation, and the concept of coset is confusing at first. After Chapter 2 the book gets easier as you go along. Indeed, after the first four chapters, the linear algebra follows easily. Finishing the chapter on linear algebra gives a basic one year undergraduate course in abstract algebra. Chapter 6 continues the material to complete a first year graduate course. Classes with little background can do the first three chapters in the first semester, and chapters 4 and 5 in the second semester. More advanced classes can do four chapters the first semester and chapters 5 and 6 the second semester. As bare as the first four chapters are, you still have to truck right along to finish them in one semester.

The presentation is compact and tightly organized, but still somewhat informal. The proofs of many of the elementary theorems are omitted. These proofs are to be provided by the professor in class or assigned as homework exercises. There is a non-trivial theorem stated without proof in Chapter 4, namely the determinant of the product is the product of the determinants. For the proper flow of the course, this theorem should be assumed there without proof. The proof is contained in Chapter 6. The Jordan form should not be considered part of Chapter 5. It is stated there only as a reference for undergraduate courses. Finally, Chapter 6 is not written primarily for reference, but as an additional chapter for more advanced courses.

This text is written with the conviction that it is more effective to teach abstract and linear algebra as one coherent discipline rather than as two separate ones. Teaching abstract algebra and linear algebra as distinct courses results in a loss of synergy and a loss of momentum. Also with this text the professor does not extract the course from the text, but rather builds the course upon it. I am convinced it is easier to build a course from a base than to extract it from a big book. Because after you extract it, you still have to build it. The bare bones nature of this book adds to its flexibility, because you can build whatever course you want around it. Basic algebra is a subject of incredible elegance and utility, but it requires a lot of organization. This book is my attempt at that organization. Every effort has been extended to make the subject move rapidly and to make the flow from one topic to the next as seamless as possible. The student has limited time during the semester for serious study, and this time should be allocated with care. The professor picks which topics to assign for serious study and which ones to "wave arms at". The goal is to stay focused and go forward, because mathematics is learned in hindsight.