The main content of linear algebra is to study the classical theory of linear relations in algebra. Because linear relationship is a simple relationship between variables, linear problems widely exist in various fields of science and technology, and some nonlinear problems can be transformed or approximately transformed into linear problems under certain conditions. The thinking method introduced by linear algebra has become an indispensable tool for scientific research and engineering application. Especially in today's rapid development and increasing popularity of computers, linear algebra, as an important basic theory course for engineering majors in colleges and universities, has a more important position and role.

Linear algebra mainly studies three kinds of objects: matrix, equation and vector. The theories of these three objects are closely related, and most of the problems are equivalent in these three theories. Therefore, when learning linear algebra, it is an important habit and quality to skillfully transfer from the narrative of one theory to another. If the matrix view is the most combined with the actual calculation, then the vector view focuses on the overall and structural considerations. So as to reveal the internal relations and essential attributes of various problems in linear algebra more deeply and thoroughly. It can be seen that as long as we master the internal relationship between matrix, equation and vector, we will be able to draw inferences from others and simplify the complex.

First, pay attention to the understanding and mastery of basic concepts, and use basic methods and basic operations correctly and skillfully.

There are many concepts in linear algebra, the important ones are:

Algebraic cofactor, adjoint matrix, inverse matrix, elementary transformation and elementary matrix, orthogonal transformation and orthogonal matrix, rank (matrix, vector group, quadratic form), equivalence (matrix, vector group), linear combination and linear representation, linear correlation and linear independence, maximal linear independence group, basic solution system and general solution, solution structure and solution space, eigenvalue and eigenvector, similarity and similarity diagonalization.

We should not only accurately grasp the connotation of concepts, but also pay attention to the differences and connections between related concepts.

There are many algorithms of linear algebra, which should be sorted out clearly and not confused. Basic operations and methods must pass the test. It is important that:

Calculation of determinant (number type, letter type), inverse matrix, rank of matrix, power of square matrix, rank of vector group irrelevant to maximum linearity, determination of linear correlation or parameters, basic solution system, general solution of nonhomogeneous linear equations, eigenvalue and eigenvector (definition method, basic solution system method of characteristic polynomial), determination and solution of similar diagonal matrix, and transformation of real symmetric matrix into diagonal matrix through orthogonal transformation.

Second, pay attention to the connection and transformation of knowledge points and network knowledge, and strive to improve the comprehensive analysis ability.

Linear algebra is criss-crossed, interlocking and interpenetrating in content, so the method of solving problems is flexible and changeable. When studying, always ask yourself if you are doing it right. One more question, okay? Only by constantly summing up and trying to figure out the internal relations among them, so that the knowledge learned can be integrated, the interface and breakthrough point can be more familiar, and the thinking will naturally be broadened.

For example, A is an m×n matrix, B is an n×s matrix, and AB = 0, then from the partitioned matrix, we can know that all column vectors of B are solutions of homogeneous equations AX = 0, and then according to the basic solution system theory and the relationship between the rank of matrix and the rank of vector group, we can have

R (b) ≤ n-r (a) means r (a)+r (b) ≤ n.

In addition, some parameters in matrix A or B can be found.

The above examples show that the knowledge points of linear algebra are inextricably linked, and algebraic problems are more comprehensive and flexible. Students should pay attention to series connection, connection and transformation when sorting out.

Third, pay attention to logic and narrative expression

Linear algebra requires more abstraction and logic. By proving the questions, we can understand the examinee's understanding and mastery of the main principles and theorems of mathematics, and examine the examinee's abstract thinking ability and logical reasoning ability. When reviewing and sorting out, we should find out the conditions for the establishment of formulas and theorems, and we should not sell ourselves short. At the same time, we should also pay attention to the accurate and concise narrative expression of language.