First, study the three basic courses as a whole, abandon isolated learning and advocate comprehensive thinking.
Engels once said: "Mathematics is the science of studying numbers and shapes." Judging from the development of modern mathematics, this philosopher's generalization of mathematics is far from accurate, but it points out the most essential research objects of mathematics, namely "number" and "shape". For example, mathematical branches such as number theory, algebra, functions and equations are all derived from the study of "number"; Mathematics branches such as geometry and topology are all derived from the study of "shape". Since the 20th century, these traditional branches of mathematics have infiltrated and crossed each other, forming the frontier research direction of modern mathematics, such as algebraic number theory, analytic number theory, algebraic geometry, differential geometry, algebraic topology, differential topology and so on. It can be said that modern mathematics is developing vigorously in the direction of the integration of various branches of mathematics.
Mathematical analysis, advanced algebra and spatial analytic geometry are just the most important basic courses of the three most important branches of mathematics-analysis, algebra and geometry. According to the characteristics of the course, the learning methods of each course are certainly different, but if you can't study and think as a whole, even if you get an A in each course, you may not be able to learn well. A senior professor in the college once complained to us: "Some problems can be solved by drawing a picture and thinking about it. Why do students now know how to do it when they bring it, and don't even draw a picture? " Of course, there must be many reasons for this deficiency. For example, from the teaching point of view, the textbooks or lectures of each course emphasize their own characteristics to a certain extent, and rarely use the holistic view to teach courses or deal with problems, and the relationship between courses is rarely involved; From the perspective of learning, most students are in an isolated learning state, that is to say, learning this course in isolation from a certain course, lacking the overall grasp and comprehensive thinking of many courses.
According to my experience, it is better to study higher algebra and spatial analytic geometry as a whole, because the core concept in higher algebra is "linear space", which is a geometric object; Moreover, many contents in higher algebra are the continuation and popularization of spatial analytic geometry. In addition, there are many analytical skills in higher algebra, such as "perturbation method", which is an analytical method that allows us to transform the problem from a general matrix to a non-homogeneous matrix. Therefore, to learn advanced algebra well, we should first jump out of advanced algebra, study the three basic courses as a whole, abandon isolated learning and advocate comprehensive thinking.
Second, correctly understand the characteristics of algebra and find the combination of abstraction and concreteness.
Algebra (including advanced algebra and abstract algebra) gives people the impression of being "abstract", which is quite different from the other two basic courses. Taking the definition of "linear space" as an example, two operations, addition and number multiplication, are defined on the set V. These two operations satisfy eight properties, so V is called linear space. I think students who study advanced algebra for the first time will find this definition too abstract. In fact, in advanced algebra, such abstract definitions abound. But this abstraction is meaningful, because we can verify that three-dimensional Euclidean space, continuous function set, polynomial set and matrix set are all linear spaces, that is to say, linear space is a concept abstracted from many concrete examples and has absolute generality. The research method of algebra is to abstract a concept from many concrete examples; Then the concept is studied by algebraic method and a general conclusion is obtained. Finally, these conclusions are returned to concrete examples and used in various ways. So "concreteness-abstraction-concreteness" is the characteristic of algebra.
After understanding the characteristics of algebra, we can learn advanced algebra with a definite aim. We can understand the abstract definition and proof through concrete examples; We can apply the conclusion of the theorem to concrete examples, so as to deepen our understanding and mastery of the theorem; We can also find and prove some new results through concrete examples. Therefore, to learn advanced algebra well, we need to correctly understand the dialectical relationship between abstraction and concreteness, and find the combination of abstraction and concreteness.
Third, advanced algebra should not only learn algebra, but also learn geometry, and build a bridge between algebra and geometry.
With the change of the times, the teaching contents and methods of advanced algebra are also developing. Before 1990s, most advanced algebra textbooks in domestic universities centered on "matrix theory", emphasizing the related skills of matrix theory. After 1990s, the textbooks of advanced algebra in domestic universities have gradually changed to "linear space theory", with more emphasis on the significance of geometry. As a microcosm, Fudan's advanced algebra textbook has also experienced such a changing process. Before 1993, the textbook adopted by Teacher Tu Bojun emphasized "matrix theory"; The textbook adopted by teacher Yao Musheng 1993 emphasizes "linear space theory". In fact, it is a global change in the teaching concept of higher algebra from simply attaching importance to algebra to paying equal attention to algebra and geometry. Perhaps this change is closely related to the development of modern mathematics!
An effective way to learn advanced algebra well should be:
Deeply understand geometric meaning and master algebraic methods.
Secondly, many problems in advanced algebra are geometric problems. We often algebra geometric problems and then solve them by algebraic methods. Of course, for some algebraic problems, we sometimes geometrically solve them and then use geometric methods.
Finally, there is a bridge between algebra and geometry, that is, the conversion language between algebra and geometry. With this bridge, we can come and go freely between algebra and geometry. Therefore, to learn advanced algebra well, we should not only learn algebra, but also learn geometry, and build a bridge between algebra and geometry.
Fourth, learn teaching materials well, make good use of teaching reference and practice basic skills.
The current textbook of advanced algebra in Fudan is Advanced Algebra (second edition) edited by teachers Yao Musheng and Wu Quanshui. This textbook has been used for nearly 20 years since 1993. The textbook is informative, focused, clear-headed, and rich in exercises. Even compared with the advanced algebra textbooks in colleges and universities all over the country, it is an outstanding work.
At present, the reference book for advanced algebra teaching in Fudan University is "Guidance on Learning Methods of Advanced Algebra (Second Edition)" edited by Yao Musheng (commonly known as "White Paper" because of its white cover). This teaching reference book is a must-have collection for undergraduates in several colleges, and its popularity can be seen.
Learning advanced algebra well and learning textbooks well are the minimum requirements. In addition, how to make good use of teaching reference books is also an important link. Many students buy reference books, mainly because some homework (including some difficult proof questions) in textbooks can be found in reference books. Of course, this is understandable, after all, this is the function of teaching reference books! However, I hope that freshmen can use reference books correctly. When you encounter problems, you should think independently first. If you really can't think of them, look at the answers in reference books to improve your ability and exercise your thinking. Note: I don't think independently, and I don't understand the answers in the reference books. I just plagiarized, which is very irresponsible to myself. I hope everyone will try to avoid it!
Finally, I want to encourage you with a poem by Mr. Hua, "Diligence is a good training, and diligence is a talent". I wish you continuous progress and academic success!