Reflections on the teaching of linear algebra: www.china-b.c0m February 2009 13 hits: 1 Source: China Paper Download Center Core Tip: [Abstract] Linear algebra is a very important course in engineering universities, and it is also a relatively abstract and difficult course to learn. According to the author's experience in theory and practice, it is proposed that the teaching of abstract concepts in linear algebra is a very important course in engineering colleges, and it is also a relatively abstract and difficult course. According to the author's experience and familiarity from both theory and practice, this paper puts forward some problems that should be paid attention to in explaining abstract concepts in the teaching of linear algebra, and explains how to carry out classroom teaching of linear algebra, which can achieve good teaching results. [Keywords:] Linear Algebra Mathematics Concept Teaching Method "Linear Algebra" is an important basic mathematics course for science and engineering majors in colleges and universities. It is not only widely used in mathematical branches such as probability and statistics, differential equations and control theory, but also its knowledge has penetrated into other disciplines of natural science, such as engineering technology, economy and social science. Moreover, this course plays an important role in improving students' mathematical literacy, practicing and improving their abstract thinking ability and logical reasoning ability. However, due to the characteristics of Linear Algebra itself, students feel abstract about its content, and it is quite difficult to understand and master the basic concepts and theories of algebra. Therefore, in order to cultivate and improve students' ability to apply mathematical knowledge and solve practical problems, it is very important to further study the teaching ideas and methods of this course and improve the teaching effect. First, strengthen the teaching and learning of basic concepts. Linear algebra is an abstract mathematical theory and method system composed of a series of basic concepts. Abstract concepts such as determinant, matrix, inverse matrix, elementary matrix, transposition, linear representation, linear correlation, eigenvalues and eigenvectors are rooted in the objective real world and have profound practical background, which is the product of relatively direct abstraction. The significance of higher mathematics and elementary mathematics and the change of thinking mode are bound to be reflected in teaching. As the continuation and perfection of middle school algebra, linear algebra is very different from it not only in content, but also in research viewpoints and methods. In the process of research, dialectical viewpoint and strict logical reasoning are repeatedly embodied, that is, the general concept is abstracted from concrete things, and then the general concept returns to concrete things. Freshmen have just entered the university, and it is difficult for their way of thinking to rise from the intuitive and concise method of elementary mathematics to the abstract and complicated way of linear algebra, so it is difficult for their way of thinking to meet the requirements of linear algebra in a short time. Most students are used to traditional formulas and are not used to understanding the essence of theorems. They use some known theorems, properties and conclusions to reason and solve problems. In concept teaching, teachers should study the characteristics and laws of the process of concept familiarity, and choose appropriate teaching methods according to the laws of students' familiarity ability development. Therefore, we should pay attention to the following points in concept teaching. 1。 Rational use of the intuition of concepts Although abstraction is a prominent feature of the course of linear algebra, intuitive teaching can also be applied to the teaching of this course and occupies an important position in teaching. Euler thought: "The science of mathematics needs both observation and experiment, and the wide application of models and graphs is such an example." Intuition contributes to the introduction and formation of concepts. For example, the concept of vector, although abstract, has a geometric intuitive background. In two-dimensional space and three-dimensional space, vectors are directed line segments, so the process from abstraction to existence can be explained from the geometric definition of vectors in teaching, which reduces the difficulty of students' abstract thinking. 2。 Make full use of the practical background of the concept and students' experience. Teachers should make full use of students' existing mathematical reality and life experience in teaching to guide and inspire students to discover and create concepts. For example, when explaining N-order determinant, we should start with the solutions of binary and ternary linear equations that students have mastered, and then find out that the solutions of equations are expressed by second-order and third-order determinants, and analyze the characteristics of second-order and third-order determinants. The second-order determinant is not difficult to see: it contains two items. If the sign is not considered, each item is the product of two elements from different rows and columns, then the question will be raised: What is the law of the sign before the right item? Similarly, the third-order determinant contains 3! =6 items, each item is also the product of three elements from different rows and columns, and contains all combinations of three elements from different rows and columns. In order to solve the n-order determinant, the concept and properties of permutation are introduced. After introducing parity arrangement, let's go back to our question. We can find that when the row labels are arranged naturally and the column labels are arranged in odd numbers, the term is negative. When the column labels are arranged in even numbers, the item is positive (problem solving). After this process, students have come into contact with and understood the N-order determinant. At this time, the definition of the N-order determinant can be given, so that students can easily understand and master the properties of the N-order determinant. 3。 Pay attention to the establishment of conceptual system. Skip pointed out: "Individual concepts must be integrated into the conceptual structure synthesized with other concepts to be effective." Concepts in mathematics are often not isolated. Understanding the relationship between concepts can not only promote the introduction of new concepts, but also help to approach the essence of learned concepts and establish the whole concept system. For example, the relationship between the rank of matrix and the rank of vector group: the rank of matrix is equal to the rank of its row vector group and the rank of its column vector group; The full rank of matrix rows (columns) is also related to the linear correlation and linearity of vector groups. Second, students should master scientific learning methods. Understanding is the key to learning. Students must memorize definitions, theorems and some conclusions on the basis of understanding and understanding its profound meaning, in order to receive ideal results. The greatest feature of linear algebra is that knowledge systems are linked one by one. The previous knowledge is the basis of the later study. For example, whether the proficiency in finding the rank of matrix by elementary transformation directly affects the rank of finding the vector group and the largest irrelevant group, and then affects the basis and dimension of the vector space generated by finding the vector group. Another example is whether the general solution of linear equations is skillful or not, which will affect the solution of the later eigenvectors, and the quadratic form will be transformed into the standard form through orthogonal transformation. Therefore, to learn linear algebra, we must adhere to new learning methods and review and consolidate them in time. Therefore, it is very necessary for teachers to review knowledge before class and for students to preview in advance. Third, strengthen students' basic practice of solving problems. A certain amount of typical exercises can help students deepen their understanding of what they have learned, cultivate their ability to solve multiple problems, reflect after solving problems, and summarize the ideas and methods of solving problems in time. There are many ways to prove the invertibility of abstract matrix. One is to define it. The second is to use the related propositions of rank. Thirdly, with the help of eigenvalue theory. The fourth is to prove that the determinant of the matrix is not zero. Fourth, cultivating and stimulating students' interest in learning is the best teacher. On the one hand, teachers are imparting knowledge, on the other hand, they should encourage students to design goals in a targeted way, so that students are willing to learn consciously and willingly. At the same time, in classroom teaching, we can choose the postgraduate entrance examination questions in recent years and some topics closely related to reality to explain or practice, so as to stimulate students' desire for learning and bring them a sense of success. In addition, some interesting application models or teaching history can be introduced appropriately to stimulate students' learning enthusiasm and improve their learning interests. Fifth, give full play to the advantages of multimedia and enhance the teaching effect. Multimedia teaching has become an important means of teaching mode in colleges and universities. Only by organically combining traditional teaching methods, teachers' own characteristics and multimedia-assisted teaching can teachers really exert the effect of multimedia classroom teaching. In short, what teachers do in teaching should not only teach students useful knowledge, but also teach students useful thinking methods and good thinking habits. Reference: [1] Zhang Xiangyang. Some experiences in the teaching of linear algebra. Journal of Shanxi University of Finance and Economics (Higher Education Edition), 2006. In Zhao Xia. Linear algebra and spatial analytic geometry. Beijing China Science and Technology Press 2003.