Elementary Algebra Brief Course. Volume 1

Author: Lan Yizhong
Publisher:
Publish Date: 2003-02-01
Features: This book is a teaching material for the "Advanced Algebra" course in comprehensive universities and normal universities. Based on over 30 years of teaching experience, the authors have systematically and comprehensively reformed the advanced algebra textbook through repeated consultations with multiple experts. This textbook has two main features: First, it closely aligns with the actual needs of classroom teaching and self-study, carefully designed with multiple levels, progressing from simple to complex, from concrete to abstract, and from vivid intuition to rational reasoning, allowing students to smoothly enter the abstract realm of algebra. The second feature is that it takes the research objects, fundamental ideas, and basic methods of algebra as the main thread of the entire book. All content, including the proofs of some fundamental theorems, is arranged according to this principle, ensuring that students receive sufficient algebraic training, achieving adequate depth and height in theory. Its scientific content meets the standards expected of an introductory modern algebra course textbook. The book is divided into two volumes, with a total of twelve chapters. The first volume (Chapters 1–5) is a fundamental textbook on linear algebra, covering topics such as vector spaces, matrices, determinants, linear spaces and linear transformations, bilinear functions, and quadratic forms. The second volume (Chapters 6–12) includes three main parts: First, metric linear spaces and Jordan canonical forms, which are more advanced topics in linear algebra; second, the ring of rational integers and polynomial rings in one and multiple variables; third, selected topics: n-dimensional affine spaces and n-dimensional projective spaces, tensor products, and exterior algebras. Each chapter of the book is accompanied by a substantial number of exercises for extracurricular practice or use in problem-solving classes. Computational problems are provided with answers at the end of the book, while more challenging problems include hints. This book can serve as a textbook or teaching reference for students in mathematics, mechanics, and applied mathematics departments of comprehensive universities and higher normal universities taking the "Advanced Algebra" course. It is also suitable for science and engineering students to read. For young teachers and mathematicians, this book is an excellent teaching reference or study material.