Knowledge points of postgraduate mathematics
The first chapter determinant
1, the definition of determinant
2. The nature of determinant
3, the value of the special determinant
4. Determinant expansion theorem
5. Calculation of abstract determinant
Chapter II Matrix
1, definition of matrix and linear operation
Step 2 increase
3. Power of matrix
4. Move the item
5. The concept and properties of inverse matrix
6. Adjoint matrix
7, block matrix and its operation
8. Elementary transformation of matrix and elementary matrix
9. Equivalence of matrices
10, the rank of the matrix
Chapter III Carrier
The concept and operation of 1, vector
2. Linear combination and linear representation of vectors
3. Equivalent vector group
4. The linear correlation of vector groups has nothing to do with linearity.
5. Ranks of Maximal Linear Independent Groups and Vector Groups
6. Inner product and Schmidt orthogonalization
7.n-dimensional vector space (math 1)
Chapter IV Linear Equations
1, Cramer's rule for linear equations
2. Criteria for homogeneous linear equations with nonzero solutions
3. Existence criteria of solutions for nonhomogeneous linear equations.
4. Structure of solutions of linear equations
Chapter V Eigenvalues and Eigenvectors of Matrices
1, the concepts and properties of eigenvalues and eigenvectors of matrices
2. The concept and properties of similarity matrix.
3. Similar diagonalization of matrices
4. Eigenvalues, eigenvectors and similar diagonal matrices of real symmetric matrices.
Chapter VI Quadratic Form
1, quadratic form and its matrix representation
2. Contract transformation and contract matrix
3. Rank of quadratic form
4. Standard form and standard form of quadratic form
5. Inertia theorem
6. Transform quadratic form into standard form by orthogonal transformation and matching method.
7. Positive definite quadratic form and its judgment
Methods of getting high marks in math review for postgraduate entrance examination
First, rationally analyze the three components and divide and rule them.
We know that the whole test paper of mathematics consists of: 82 points for high number+34 points for line generation+34 points for probability theory; It is obvious that calculus accounts for the vast majority; In addition, many problems in probability theory need calculus tools. In fact, the score of calculus is higher than 82, which should be about 100. Therefore, students must lay a good foundation of calculus when reviewing in the early stage; Linear algebra is more difficult, after all, there is not much content. Moreover, the essential ideas of matrix, vector, linear equations, eigenvalue and eigenvalue, quadratic form are all consistent. The basic tool used is to do the elementary transformation of matrix and find the structure of the solution of linear equations. The difficulty of line substitution is that the basic ideas of each part are the same, but they are different concepts. In this way, the relationship between chapters is particularly close, and the logical relationship is tight: for example, a problem unrelated to linear correlation is essentially the same as whether a homogeneous equation has a non-zero solution; Some proofs of vector linear correlation and irrelevance can be simply completed by the solution of linear equations; It is precisely because of the great internal relevance of knowledge points that the examination difficulty of linear algebra is improved. However, because there are not many knowledge points in linear algebra itself, as long as each part is proficient to a certain extent and deeply understood, it is natural to master the connection and logic.
In the third part, many basic concepts of probability theory have actually been touched by us in high school, and we have also learned some simple calculations of event probability and basic probability. Generally speaking, probability theory is the simplest of the three parts. Not only is there less content, but the questions in the annual exam are also particularly fixed. I really think this part can be completed by surprise. To sum up: calculus is the difficulty and focus of the whole postgraduate entrance examination. We must lay a solid foundation; Linear algebra is a difficult point, which can be broken through proficiency and thinking; Probability theory, as long as your previous knowledge is solid enough, there is no problem at all. In addition, during the review process, many students asked me if I wanted to watch calculus, linear algebra and probability theory at the same time. My suggestion is: a point to work together, a point decomposition! Modest and prudent, neither arrogant nor impetuous.
Second, focus on energy and choose teaching AIDS.
There is a phenomenon every year, that is, choosing teaching AIDS, experience stickers, brothers and sisters, and so-called better materials mentioned in various channels are eager to buy, but how many people can finish reading them after buying them? I have to remind you here: you should know that the math test for postgraduate entrance examination is depth, not breadth; I always thought three sets of books were enough:
(1) Textbook, Advanced Tongji Edition; The fifth edition of line generation statistics; Probability theory, fourth edition of Zhejiang University;
But I have to remind you here that it takes a lot of time and energy to master these four books thoroughly; There are many things in it that are not tested, even in the syllabus. In fact, when reviewing, many students put too much energy into those subjects that are not tested and are relatively biased. This will lead to a lot of energy waste. To this end, when I teach mathematics, I will preview the outline in advance, which ones should be tested and which ones should not be tested; Practice what to do and what not to do after class. So that everyone can concentrate.
(2) True questions
In any case, every exercise refers to many prototypes of real questions, and even directly refers to real questions. Needless to say, the value of the real question. But every student handles it very simply. As long as you do it right, you will pass. Don't look back and think about whether your practice or your thinking meets the requirements of the proposer. As for the real questions, it is more appropriate to do them for about 5 times for the better typical questions. For some very routine questions, you can do it 2-3 times. In short, we must thoroughly study the real questions and maximize the value of the real questions. My suggestion: There are many teaching AIDS on the market. I think it is enough to choose what is recognized by everyone, give full play to its value and study it carefully. Don't follow the trend and buy too many teaching AIDS, which will distract you, but it doesn't meet the depth and difficulty of the postgraduate entrance examination.
Third, master the correct review method: killing people and killing their hearts.
When reviewing mathematics, it is true that everyone has their own ideas, but remember, it doesn't matter what you think, the key is what the proposer thinks. Especially when doing problems, don't just take it as a standard. Be sure to analyze the knowledge points and test the logic behind them. Finally, I must ask myself whether this method is the one that the proponent wants me to use. What are the shortcomings and what are the neglected details, we must take a good look. In addition, the characteristics of mathematics examination: learn to think rather than learn to do problems, but it is difficult to generate ideas before we are familiar with a problem; Therefore, in the whole review process, I always ask students to be familiar with it first, and then they must really turn this question into their own through their own thinking, so as to draw inferences from others and constantly cope with changes. In addition, students are prone to two misunderstandings when doing problems:
1, as soon as it comes up. Students who have done real questions will find that the setting of many questions is very skillful; This skill is not opportunistic, but requires you to be familiar with the knowledge points and think it over before you can come up with it. I remember that in the exams in recent years, I worked out many answers to 10, 1 1 in less than a minute. Of course, many students may not believe it. I will show it to my classmates in class myself. I'm not saying that I'm fine, but when you are skilled enough, you will talk to the suitor, and I feel the harmonious heartbeat of the sacred unicorn. So when you do the problem, you must look, think and do it.
2. Deliberately remember some clever methods. In mathematics for postgraduate entrance examination, I always think that the best way is definitely not opportunistic, but natural. For example, Fermat's Lemma may not be directly tested, but it proves that the thoughts and thinking you use are necessary for the postgraduate entrance examination. Therefore, we must master its proof carefully.
Guide to reviewing mathematics for postgraduate entrance examination
1. Think about doing the problem and summarize.
Many students are so confused. There are still many problems that they have done but can't do. The most exasperating thing is that many questions are clearly done, and they can't be done when we meet again! This is a common problem of many students, and I don't want to solve it. I always feel that I can't do it, just look at the answer, and I won't seriously think about why I can't, what is the problem-solving skill, whether I can do the same type of problem as it, and so on. In fact, these are very important, reminding everyone to learn to think, learn to "remember", and the most important thing is to draw inferences from others, so as to get rid of the ups and downs of the sea of questions, effectively solve the questions and improve them efficiently!
2. Pay attention to the foundation and cultivate reverse thinking.
Many times, candidates will fall into the ocean of blind questions, which is also the reason why many candidates feel headache about mathematics. In fact, when reviewing knowledge points in the early stage, we should focus on the derivation of definitions and theorems, attach importance to the methods and skills in derivation and examples, carefully analyze these methods and apply them to corresponding exercises, which is much more efficient than doing a lot of repeated exercises.
At the same time, thinking habits greatly affect the learning effect. When entering the math review for postgraduate entrance examination, most people inherit the habit of studying in the past, and their thinking is basically stereotyped, that is, they enter the mindset. Habitual thinking has advantages on the one hand, and restricts the improvement of academic performance on the other. What we have to do now is to break the inertia thinking!
3. Do the problem from beginning to end to improve the calculation ability.
Mathematics is not equal to doing problems, but learning mathematics well is bound to do problems. So how do you do the problem? We say that a solid foundation is fundamental, and then do the problem on this basis. At the same time, I remind you that you must form a good habit of reviewing and work out the math problems you get from beginning to end. This is a kind of training of calculation ability, especially when the amount of calculation is large. Without the usual training, it is difficult to have enough spare capacity in a short time in the actual exam.
4. Think deeply and be good at summing up
Examination is not only to examine our basic concepts, theories and methods, but also to examine our ability to use knowledge flexibly. Therefore, it is difficult to summarize the characteristics of this test proposition only by relying on textbooks. Therefore, to understand the exam, the real questions of the exams over the years are students who are preparing for the postgraduate entrance examination.
When you choose real questions, you should consider whether you can help us really sum up these problems through the analysis of real questions. For each question, how should we analyze and discuss it? Are there any possible changes in the process of analysis and discussion? What have been tested so far? Those are what we should pay attention to in the next review. It is even more important for you to summarize each part in this way or to help you summarize it through such related counseling books.
5. Try to find out the real problem and grasp the direction.
The role of real questions can not be ignored. After more than ten years of examination, quite a number of topic patterns have been decided, and many postgraduate entrance examination questions are similar. After tempering, the real questions of postgraduate entrance examination have high reference value in thought and need more attempts. Especially in the past two years, the examination questions reflect the way and thinking of the proposer, which we should pay more attention to. So students must pay attention to the real questions!
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