Linear algebra is mainly matrix operation and proof. The most important thing is to understand the definition in depth, and it is best to explain the principle to others. Master the definition, calculate carefully, and of course do some exercises.
With regard to the review of math, especially linear algebra, this paper puts forward a four-word strategy of "early", "outline", "basic" and "lively" for the reference of science and engineering and economic candidates.
First, "early". Advocating the word "early" is to remind candidates to plan, arrange and start early, because mathematics is a discipline with rigorous thinking, strong logic and relatively abstract. Unlike some subjects with more memories, mathematics needs to understand many concepts and flexible methods, and understanding concepts, especially abstract concepts, is a gradual process that requires thinking and thinking. We need to think deeply and study deeply from different angles and sides. In short, it takes time, and any idea of making a surprise attack or plane crash is not desirable, which is impossible for most candidates to succeed. On the other hand, early planning, early arrangement and early start are the policy of "stupid birds fly first", which is required by the fierce competition in postgraduate entrance examination. Prepare one day earlier, get more grades, and get more assurance. Now many freshmen and sophomores are already preparing for the postgraduate entrance examination two to three years later. It seems a little early, but as a goal and pursuit, it is beyond reproach.
Second, the "outline". Highlighting the syllabus means studying the syllabus carefully, reviewing and preparing for the exam in a planned, serious, comprehensive and systematic way according to the exam contents, requirements and sample questions stipulated in the syllabus, and strengthening the pertinence of preparing for the exam.
Because the textbooks of basic mathematics in China (advanced mathematics, linear algebra, probability theory and mathematical statistics) are not uniform, schools and majors have different requirements for these courses. The Ministry of Education has not designated uniform textbooks or reference books as the basis for proposition. Instead, it takes the Mathematics Examination Outline of the National Postgraduates' Unified Entrance Examination (hereinafter referred to as the Outline) formulated by the Ministry of Education as the normative document for the examination and makes the proposition based on the outline.
In order to let the majority of candidates have a certain understanding of "what to test" (not blindly preparing for the exam), the examination questions ordered by the examination center of the Ministry of Education every year have the characteristics of stability and continuity. The sample questions and previous questions provided by the outline are also to let candidates know what to test. In the previous test questions, there have never been any questions that are off topic, strange questions or beyond the scope of the outline. Of course, a good test question must first be. The total number of test questions must be limited, and the coverage of test questions should be as large as possible. Therefore, it is not advisable to blindly do difficult questions, even strange questions or digressions. "sea tactics" cannot replace comprehensive and systematic review. Because the coverage of the test questions is very large, almost all chapters are covered every year, so it is not appropriate to ignore some contents. Any "guessing" and luck will lead to failure. Only according to the outline.
Please note that there are some changes in this year's syllabus: all approximate calculations have been cancelled, especially the word "preliminary" has been cancelled in the second math test paper "Preliminary Linear Algebra", and the chapter "Eigenvalues and Eigenvectors" has been added.
Third, the "foundation". Emphasizing the word "foundation" means emphasizing three foundations in mathematics learning, namely, attaching importance to the understanding of basic concepts, mastering basic methods and mastering basic operations.
Incomplete understanding of basic concepts will bring difficulties and confusion to solving problems. Therefore, we must understand its connotation, study its extension, understand its positive significance, and think and understand its side and negative side. For example, regarding the rank of a matrix, the textbook definition is: A is a negative Xn matrix. If there is a sub-formula of order R in A that is not zero, all sub-formulas above order R (if it still exists) are zero. Write rank (a): r (or r (a) = r, rank a = r). Obviously, the key points of the definition are as follows: 1 has at least one sub-formula of order r. A is not zero; 2. All r orders and above are zero. 3. If all sub-formulas of r+ 1 are zero, all sub-formulas of order r and above must be zero. Point 2 and point 3 are equivalent conditions. Can the r-order subformula be zero? Can the formula less than r order be zero? Can all r- 1 sub-formulas be all zero? These are extensions of the concept of hierarchy, if these concepts are clear. Then the following multiple-choice questions are solved.
Example 1 let a be an m×n matrix and R (A) = R
(b) There is an R-order formula that is not equal to zero, and there is no r+ 1 order formula that is not equal to zero.
(c) A formula of order R is equal to zero, and a formula of order r+ 1 is not equal to zero.
(d) Any sub-formula of order R is not equal to zero, and any sub-formula of order r+ 1 is equal to zero.
A: (b)
Mastering basic methods is not the same as rote learning. On the contrary, we should grasp the essence of the problem and remember it properly on the basis of understanding. To minimize what needs to be memorized, many methods can be memorized through practice. For example, a real symmetric matrix must have an orthogonal matrix, and the orthogonal matrix is transformed into a diagonal matrix through orthogonal transformation. There are many steps, but it is not difficult to solve them through practice.
You should be proficient in basic calculations. Learning mathematics is inseparable from calculation, and it is necessary to be proficient in calculation. Of course, you have to do a certain amount of exercises. Through a certain amount of practice, you should practice the basic skills of calculation. In the process of practice, we should consciously improve our calculation ability and accuracy, and develop good calculation habits and scientific style. Especially in linear algebra, the operation is not complicated, and a lot of operations are addition and multiplication that everyone has been proficient in for a long time, so it seems that you can develop good calculation habits and scientific style. For example, in the first four chapters of linear algebra (determinant, matrix, vector, equation), most operations are elementary transformations. Using elementary transformation to find the value of determinant, inverse matrix, rank of vector group (or matrix), maximum linearly independent group of vector group, solution of equation, etc. It is conceivable that once there is a numerical calculation error in the process of elementary transformation, what will your answer be? Judging from the previous mathematics test questions, the content that needs to be calculated and scored every year is about 70%, which shows the importance of cultivating computing ability. Just listening (listening to various remedial classes) without practicing, just watching (reading various counseling materials) without practicing, looking for problems to do, is not suitable for ordinary candidates. Among the previous candidates, there are many who have had painful lessons.
Fourth, "live". Linear algebra is characterized by many concepts, theorems, symbols and operation rules, criss-crossing contents and close knowledge connection. Therefore, candidates should fully understand the concept, grasp the conditions, conclusions and applications of theorems, be familiar with the meaning of symbols, master various operation rules and calculation methods, sum up in time, grasp the relations and laws, and simplify the complex.
The contents of each chapter of linear algebra are not isolated, but mutually infiltrated and closely related. For example, a is an n-order square matrix, and if | A | ≠ 0 (A is called an odd matrix). < = >； A is an invertible matrix.
Example 2 (ninth problem of mathematics in 20065438+0) Let α 1, α2, …, αs be linear equations AX=0, β1= t1+T2 α 2, β 2 = t/kloc-0.
The key points to solve this problem are: (1) The solution of any t 1, t2, βi, i = 1, 2, …, s is still ax = 0；; (2) For any t 1, t2, β 1, β2, …, βs vector number is s; (3)β 1, β2, …, βs, linearly independent.
Variable (1) (changed to linear correlation test)
It is known that the vector groups α 1, α2, …, αs are linearly independent, β1= t1α1+T2 α 2, β2 = t 1α 2+t2α 3, …, β s = t/kl.
Variant (2) (Test Problem of Changing the Level of Vector Group)
It is known that the ranks of vector groups α 1, α2, …, αs are S.β1= t1α/+t2α 2, β2 = t 1α 2+t2α 3, …, β s = t/.
Variant (3) (the test questions are changed to equivalent vector groups)
It is known that α 1, α2, …, αs is linearly independent, β1= t1α1+t2α 2, β2 = t 1α 2+t2α 3, …, β s = t/kloc.
Variant (4) (The test questions are changed to the basis of subspace)
Let y be the subspace of Rn, α 1, α2, …, αs be the basis of V, β1= t/α1+T2α 2, β2 = t 1α 2+t2α 3, …, β.
Don't you think the above variants are basically the same? What are the main points of their answers?
Changing the difficulty of the test questions and concretizing the number of vector s became the twelfth question in the second math test paper of 200/kloc-0.
Variant (5) It is known that α 1, α2, α3, α4 are linear equations AX=0, β1= t1+t2α 2, β2 = T 1α 2+T2α 3.
Can't you "do whatever you want" by changing the parameters?
Variant (6) it is known that α 1, α2, …, αs is AX = 0, β 1 = t 1+t2α 2, β2 = t 1α 2+t2α 3, …, βs = t65438.
If you don't realize that the above variants are essentially the same, then you haven't studied "living" linear algebra, and your knowledge is still isolated.
Due to the close connection and infiltration of knowledge, comprehensive examination questions are no longer attached to a chapter or a section (the exercises attached to a chapter or a section actually provide the problem solver with tips for using the contents and methods of the chapter or section to solve problems), which will bring difficulties to candidates. It is not easy to learn "live", so we should always sum up and broaden our thinking.
Example 3 shows that A is an n-order positive definite matrix and B is an n-order antisymmetric matrix, and proves that A-B2 is a positive definite matrix.
The analysis of this topic itself is very enlightening. What is known is a positive definite matrix, and what is to be proved is also a positive definite matrix, which obviously belongs to the positive definite test questions in quadratic form. The specific answer is as follows.
B is an antisymmetric matrix, so Bt =-B.
Let X≠0, because a is positive definite, so xtax >；; O, and XT (B2) x = xtbtbx = (bx) tbx ≥ 0.
Therefore, XT (a-B2) x = XT (a+(-b) b) x = XT (a+BTB) x = xtax+(bx) TBX > O.
So A-B2 is a positive definite matrix.
Variant (1) is known that A is an n-order positive definite matrix and B is an n-order antisymmetric matrix, which proves that A-B2 is reversible. This variant of V requires that A-B2 is reversible, but it is known that A is positive definite. In order to make use of the known conditions, we can also think about whether A-B2 is positive definite, that is, if A-B2 is proved to be positive definite, then A-B2 is naturally proved to be reversible.
Variant (2) shows that B is an n-order antisymmetric matrix and E is an n-order identity matrix, which proves that E-B2 is reversible.
In this variant, the condition that A is a positive definite matrix is hidden, but the specific positive definite matrix E is given. It is very difficult to prove the reversibility of E-B2 from the perspective of proving positive definite, which requires the accumulation and summary of experience.
Because of the extensive contact and mutual penetration of knowledge, it creates conditions for solving multiple problems with one problem. We can study the problem from different angles, find a suitable starting point, and finally find the answer to the problem.
In short, only by attaching importance to the three basics, the relationship between chapters, the study of test questions from multiple angles, flexibility and comprehensiveness, and application can we achieve ideal results.
In fact, I personally think that among the three parts of high number, line generation and probability, line generation is the simplest and less flexible than high number. As long as you master the basic knowledge, do more questions and work hard, it is easy to get high marks in this part.