1. Linear algebra is a branch of mathematics, and its research objects are vectors, vector spaces (or linear spaces), linear transformations and linear equations with finite dimensions. Vector space is an important subject in modern mathematics. Therefore, linear algebra is widely used in abstract algebra and functional analysis; Through analytic geometry, linear algebra can be expressed concretely. The theory of linear algebra has been extended to operator theory. Because the nonlinear model in scientific research can usually be approximated as a linear model, linear algebra is widely used in natural science and social science. Linear algebra is an important content of mathematics courses for science, engineering and economic management. The proportion in the postgraduate entrance examination generally accounts for about 22%. \x0d\ II。 Skills and methods \x0d\ 1. Pay attention to understand and master basic concepts, and use basic methods and operations correctly and skillfully. There are many concepts of \x0d\ linear algebra, the important ones are: \x0d\ 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, \ \x0d\ such as matrix a = (α 1, α2,? , αm) and b = (β 1, β2? , βm) is equivalent, that is to say, B can be obtained from A through elementary transformation. To do this, the key is to see whether the rank r(A) and r(B) are equal, and the vector group α 1, α2,? αm and β 1, β2,? βm equivalence means that two vector groups can be expressed linearly with each other, so they have the same rank. However, when the vector groups have the same rank, there is no guarantee that they will be linearly represented with each other, and the information of vector group equivalence cannot be obtained. Therefore, the vector groups α 1, α2,? αm and β 1, β2,? βm is equivalent, and the matrix A = (α 1, α2,? αm) and b = (β 1, β2,? βm) is equivalent, but the equivalence of matrices A and B does not guarantee the equivalence of these two vector groups. \x0d\ Another example is that the real symmetric matrices A and B contract, that is, there is a reversible matrix C that makes CTAC = B. To achieve this, the key is whether the positive and negative inertia indexes of the quadratic xTAx and XTXBX are the same, and the similarity between A and B means that there is a reversible matrix P that makes P- 1ap = B, and then we know that A and B have the same eigenvalue. If the eigenvalues are the same, we can know the positive and negative inertia indexes. \x0d\ There are many algorithms in linear algebra, which should be sorted out to avoid confusion. Basic operations and methods must pass the test. It is important to calculate determinant (number type and letter type), find the inverse matrix, find the rank of matrix, find the power of square matrix, find the group whose rank has nothing to do with maximum linearity, find the judgment or parameters related to linearity, find the basic solution system and find non-homogeneous. Basic solution system method of characteristic polynomial), judge and find similar diagonal matrix, and transform real symmetric matrix into diagonal matrix through orthogonal transformation (that is, transform quadratic form into standard form through orthogonal transformation). \x0d\ 2。 Pay attention to the connection and transformation of knowledge points and network knowledge, and strive to improve the comprehensive analysis ability. \x0d\ Linear algebra is criss-crossed, interlocking and interpenetrating in content, so the method of solving problems is flexible and changeable. When reviewing, you should always ask yourself if you have done 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. \x0d\ If A is an m×n matrix, B is an n×s matrix, and AB = 0, then we can know that all column vectors of B are solutions of the homogeneous equation AX = 0 by using the block matrix, and then according to the theory of basic solution system and the relationship between the rank of matrix and the rank of vector group, we can have \x0d\ r(B)≤n-r(A). Then, if P- 1ap = ∧, we can know that A has n linearly independent eigenvectors, and P is composed of linearly independent eigenvectors of A, then we can know from the relationship between eigenvectors and the basic solution system that if λi is the multiple eigenvalue of ni, then the homogeneous equation (λ IE-A) x = 0. Furthermore, the rank r (λie-a) = n-ni is known. Then, if A cannot be diagonalized similarly, the eigenvalue of A must have multiple roots and the eigenvalue λi makes the rank r (λ ie-a) < n-ni. If a is a real symmetric matrix, it is known that for each eigenvalue λ i, there must be r (λ ie-a) = n-ni. \x0d\ For example, for the determinant of order n, we know: \x0d\ If | a | = 0, ax = 0 must have a non-zero solution, while ax = b has no unique solution (there may be infinite solutions or no solutions). When | a | ≠ 0, the unique solution of ax = b can be found by Clem's rule; \x0d\ Prove whether matrix A is invertible by | a |. If it is invertible, use adjoint matrix to find a-1; \x0d\ For n n n-dimensional vectors α 1, α2,? αn can be determined by determinant | a | =| α 1 α 2? Judging the linear correlation of vector groups by whether α n | is zero; \x0d\ The rank r(A) of matrix A is defined by the highest order number of non-zero terms in A. If r (a) < r, all terms of r in A are 0; \x0d\ Find the eigenvalue of matrix A, and you can calculate determinant | λ e-a |. If λ = λ 0 is the eigenvalue of a, then the determinant | λ 0e-a | = 0; \x0d\ To judge the positive definiteness of quadratic xTAx, the principal components of the available order are all greater than zero. \x0d\ These are precisely because the knowledge points of linear algebra are inextricably linked, so algebra problems are more comprehensive and flexible. Students should pay attention to series connection, connection and transformation when sorting out. \x0d\ 3。 Pay attention to logic and narrative expression \x0d\ 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.