2008-Postgraduate Entrance Exam Mathematics-Linear Algebra-(Volume II)

Author: You Chengye
Publisher:
Publish Date: 2007-02-01
Features: Among the three courses covered in the postgraduate entrance examination mathematics exam, linear algebra is the most conceptual one. The depth of understanding of algebraic theory directly affects the ability to tackle algebraic problems during the exam. The preparation for linear algebra should always prioritize deepening the understanding of theory. Conceptual and proof problems in algebra are often challenging for candidates. The ability to solve such problems directly reflects the candidate's level of understanding of algebraic theory. While there are not many types of computational problems in linear algebra, and the methods are quite elementary, the computational workload is often large. To excel in algebraic computational problems, one must be proficient and clever. "Proficiency" refers to mastering the computational methods of various problem types and understanding the theoretical principles behind them. "Cleverness" refers to having a concise and clear problem-solving approach. Cleverness can make you see the bigger picture and find the best solution path, thereby reducing computational workload and achieving the dual benefits of saving time and reducing errors. Achieving "cleverness" also requires a good understanding of theory. In summary, mastering algebraic computational problems also requires a clear understanding of theory. From a theoretical perspective, algebra is also a relatively difficult course. Many of its concepts and properties are complex and abstract, especially the close connections between different parts of the content, which are often overlooked by many candidates in their past studies. Based on the above reasons, the author has carefully designed the concept review section when writing this book. Although this section is not the main part of the book in terms of, it embodies the author's many years of teaching experience in linear algebra and their unique understanding of the course. On this basis, it aims to provide candidates with a systematic and organically connected algebra review material that is easier to understand, remember, and use. Readers will find that the concept review section of this book is not just "explanations of terms" from the exam syllabus. The exam syllabus is naturally an important basis for writing this book, but we do not strictly "adhere" to it. Some content goes "beyond" the syllabus. Readers will also find that the content of this book is not a simple condensation of general textbooks. It differs from general textbooks in its system and highlights the connections between different parts of the content. In terms of teaching methods, it also has its own characteristics. Here are a few points to consider.
(1) Some candidates think, "If it's not mentioned in the exam syllabus, it won't be tested, so there's no need to review." This view is one-sided. Mathematics has strong systematic properties, and linear algebra is particularly so. Some content may not be listed in the syllabus requirements, which only means it won't appear directly as an exam question. It still needs to be reviewed because it is often indispensable for understanding the overall theory.
(2) The ultimate goal of review is to pass the exam. With the increasing competitiveness of postgraduate entrance exams, the form and difficulty of exam questions are changing, and most are no longer the basic types commonly found in general textbooks. These questions not only require candidates to be proficient in solving computational problems but also to have a good understanding of related concepts and properties. In this book, we introduce some content not covered in general textbooks and teach practical and concise methods for common problems. These methods do not involve advanced theoretical knowledge but are just a small step beyond the exam syllabus, making them easy to understand.
(3) During the review stage, attention should be paid to the connections between different parts of the content, which is also a focus of this book. These connections are not only reflected directly in the content but also considered in the arrangement. Several fundamental concepts in algebra are not difficult to understand, and most students who have studied them will not forget them. We have gathered these basic concepts at the beginning of the book and provided a brief introduction, allowing candidates to recall the fundamental concepts of algebra at the start of their review before focusing on the more challenging parts. At the same time, this is also done to emphasize the horizontal connections of concepts when explaining each chapter, without being limited by the order in which the concepts appear.
Examples are the main part of the book's content. In each chapter, we have selected a rich variety of examples (some of which are more difficult exam questions from previous years), covering all typical problems related to the content. For problem-solving methods, we do not aim for comprehensiveness or introduce cumbersome and unessential methods. Instead, we strive for conciseness, natural reasoning, and inspiration. When necessary, we will also use notes to emphasize the thought process and methods in problem-solving. At the end of some examples, we provide related problem types for readers to practice immediately, achieving the effect of learning by doing. The examples include proof problems, some of which are quite challenging. In the actual exam, such difficult questions may not be as common, but by analyzing and proving these examples, readers can grasp the thought process and key points of problem-solving, improving their own skills. The book also includes a wide range of practice problems. The examples and practice problems can be said to cover almost all types of problems in this course. The exam syllabi for the first, second, third, and fourth mathematics tests are almost the same in linear algebra (with the first test adding a vector space section, see Section 3.8). Therefore, this book is suitable for all types of candidates. Due to the urgency of time, this book may inevitably have shortcomings, and we welcome readers to provide valuable suggestions and opinions.