Second, skills and methods
1, pay attention to understanding and mastering 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.
In previous years, candidates often did not accurately grasp the connotation of concepts, nor did they pay attention to the differences and connections between related concepts, which led to mistakes in doing the questions.
For example, the matrix A = (α 1, α2, …, αm) is equivalent to B = (β 1, β2…, βm), which means that B can be obtained from A through elementary transformation. To do this, the key is to see whether the ranks r(A) and r(B) are equal, and the vector group α 655. Therefore, they have the same rank, but when the vector groups have the same rank, they cannot be guaranteed to be linear with each other, and the information of vector group equivalence cannot be obtained. Therefore, from the equivalence of vector group α 1, α2, ... α m and β 1, β2, ... β m, we can know that the matrix A = (α 1, α2,
Another example is the contraction of real symmetric matrices A and B, 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 quadratic xTAx and XTXB 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 eigenvalue is the same, we can know that the positive and negative inertia indexes are the same, but the positive and negative inertia indexes are the same.
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.
2, pay attention to the connection and transformation of knowledge points, knowledge should be networked, 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 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.
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.
For another example, if A is an N-order matrix, it can also be diagonalized. Then if P- 1ap = ∧ is processed by block matrix, it can be known that A has n linearly independent eigenvectors, and P is composed of linear independent eigenvectors of A. From the relationship between eigenvectors and basic solution system, it can be known that if λi is the multiple eigenvalue of ni, then the homogeneous equation (λ IE-A). 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.
For another example, for the determinant of order n, we know that:
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 Cramer's law.
| a | can be used to prove whether the matrix A is invertible. If it is invertible, a-1can be found by the adjoint matrix.
For n N-dimensional vectors α 1, α2, ... α n, whether determinant | a | = | α 65438+α 2...α n | is zero can be used to judge the linear correlation of vector groups.
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;
The eigenvalue of matrix A can be obtained by calculating determinant | λ e-a |. If λ = λ 0 is the eigenvalue of a, then the determinant | λ 0e-a | = 0;
To judge the positive definiteness of quadratic xTAx, we can use that the principal components of the sequence are all greater than zero.
All this is precisely because the knowledge points of linear algebra are inextricably linked, and algebra problems are more comprehensive and flexible. Students should pay attention to series connection, connection and transformation when sorting out.
3. Pay attention to logic and narrative.
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.