After a low-level scientific research dog left the study room with a sense of surprise of his appearance and class division in his school, Jiang Shuiyuan continued his book-making journey. As for Deputy Director Jing Peng's words, he felt that he could only listen to half of them at most, that is, various textbooks should be compared and looked at each other, and the key was to see how their terms were defined and how the theorems were expressed, and to think about the logic behind each book, and find commonalities and characteristics from them.

Of course, this learning method can only be used by Jiangshuiyuan. If others dare to brush this way, their heads would have been covered in a pile of mud!

As for the half that Jing Peng said, "Advanced Algebra will not be too difficult", it is like the adults say to the child, "I will collect your New Year's money for you and give it to you in the future." Just listen to it, whoever believes is stupid.

In the next week, Jiang Shuiyuan lived a very peaceful life, just reading books and writing questions. During this period, he only received a letter from Mr. Qiao Zhizhi. He didn't say anything else, but only looked at the old man's thin and neat small calligraphy. This made Jiang Shuiyuan very ashamed and secretly decided to practice calligraphy when he had time.

In the letter, Mr. Qiao said that after he went back, he thought about it and thought that Xiao Jiang was able to master the thirteen classics and twenty-four histories and other major works. He discussed it with his colleagues and formulated 6 major directions and 13 small questions. He hoped that Xiao Jiang could think more and write more. If he had any experiences, he could write them down and send them to him directly. The words were not limited to length, and three or two thousand words were not limited to less than 100,000 or 80,000 or more.

I took a rough look at the 6 major directions of the river. From the pre-Qin philosophers' thoughts on governing the country, the sacrificial system of the Han Dynasty, to the interaction of Confucianism, Buddhism and Taoism in the Wei, Jin, Southern and Northern Dynasties, the transformation of social order in the Tang and Song dynasties, and then to the development of Neo-Confucianism in the Song and Ming dynasties, the evolution of academic thought in the Qing Dynasty, almost all major dynasties in history, fully reflecting the characteristics of the integration of classics and history. The key is that the topic is big enough, let alone practicing writing for college students, it is enough to make a dissertation for doctoral students!

As for the 13 small questions, the content is relatively specific, at least after reading it, people know what to do about it. But what about listing 10 or 20 reference books at the end of each question? No matter what, no matter how you prepare for the exam for advanced algebra!

Matrix, determinant, linear space, polynomial ring, tensor product, I'm here!

A week passed in a blink of an eye, and Jiang Shuiyuan faced a quiet look and welcomed the second exam for admission. In fact, he was already panicked: the boss who set the questions must be proud. Even if the exam is deeper, more knowledge points, and more calculations, I have recognized it. Don’t use those stupid skills anymore, as it will be really dead!

The questions were given this time, Director Qiu and the low-level scientific research dog, who had sparsely sent a question, but Director Shen did not come. Deputy Director Jing Peng replaced him.

Qiu Wanping shook his brighter head and announced: "The content of this test is advanced algebra, with a total of 20 questions. Answering 12 correctly is qualified, and answering 17 correctly is excellent. The time is one day, that is, the paper is handed in before 8:30 tomorrow morning. The test form is open paper. Are there any questions?"

Advanced algebra needs to study freshman year and next two semesters, which is the so-called "higher age," generally has 10 credits. There are 20 questions in the exam, and it is still 2 questions and 1 credit. It looks quite fair, just like last time. However, when it was heard that it was one day, Jiang Shuiyuan shook his hand: I go, it seems that the question is not simple, maybe I have to stay up late again!

Seeing that Jiang Shuiyuan had no objection, Qiu Wanping signaled that the low-level scientific research dogs should be sent to the paper.

When Jiang Shuiyuan got the question, he immediately eliminated all distracting thoughts and started answering the question in a race against time. The first question was to find the eigenvalues ​​and eigenvectors of the matrix, which belonged to the basic operation in advanced algebra. He picked up the pen and divided it into two in three or five minutes, and easily calculated the answer in less than five minutes. After calculating, he realized something was wrong: Hey, how can this question be so simple? How can it be so simple?

Did I step on the mine?

He was so scared that he quickly turned around and carefully checked from beginning to end, but no problems were found. So he raised his head and looked at Qiu Wanping with a cute look: What's going on? Is it because you found out your conscience or you got the wrong test paper? Why is today's test paper so simple? Honestly speaking, is there any conspiracy?

Qiu Wan flatly looked at him: "Do your questions well and don't look around!"

"Didn't you say it's open book?"

"Uhhh!" Qiu Wanping was choked and glared at Jiang Shuiyuan and Jing Peng in anger, then left with a look.

Why do you glar at me? Is it so difficult for menopause elderly people to do? Jiang Shuiyuan shrugged and lowered his head to continue to do the second question. The second question is to find the normative basis of a homogeneous linear equation system, which is also a routine operation. After five minutes of doing it, he was completely shocked: Why? Why? Is it that my sincerity moved God and made my dreams come true?

Thank you God, thank you Earth, thank you fate for letting us meet!

No, no, "Even if we have achieved extremely great results in our work, there is no reason to be proud and arrogant. Humility makes people progress, and pride makes people fall behind. We should always remember this truth." Jiang Shuiyuan silently recited the librarian's motto three times, corrected his mindset, and continued to do the questions.

Before ten o'clock, Jiang Shuiyuan had already solved all the previous twelve questions like chopping melons and vegetables, and successfully reached the passing line. He shook his wrists and decided to do everything in one go, ending all the battles before lunch, and using the ham fried rice from Xuesan Canteen to celebrate the successful completion of the 10 credits in the higher generation. Then he encountered question 13.

Suppose that it is an n-order real symmetrical matrix and 0. Prove that: If it is a semi-positive definite matrix, there is an orthogonal matrix, so that&039;&p;b;1,,&p;b;r,0,&039;0,,0,&p;r1,,&p;p;p;.

The question looks a bit difficult, and the key is also a tail after the key: at least two methods are used to prove it. From this point of view, the questioner is not changing his mind, putting down his butcher's knife and eating vegetarian food and chanting Buddha's name, but his thinking has changed slightly. The first 12 questions are basic questions, which ensures that as long as the basic skills are solid, the next 8 questions are improved. The difficulty is quickly raised by an order of magnitude. It seems that it is deliberately to screen and grade the answerer's abilities.

Since that's the case, then come on!

Jiang Shuiyuan made a brief calculation in his mind. Although the inverse exchange matrix does not necessarily have a common eigenvector, the semi-positive qualitativeness only requires limiting the problem to the characteristic subspace of zero eigenvalues ​​to discuss. At this time, the inverse exchange becomes exchange. Therefore, the first method is to use the invariant subspace theory to transform the problem into geometric language, which can easily prove the conclusion.

Just after he finished writing, he thought of another method, using the orthogonal similarity standard shape theory of real symmetric arrays, assuming that it is orthogonal similarity standard shape to do it. Well, ss

Progress 13/20

Just as Jiangshuiyuan was about to turn to the next question, he suddenly thought of a more clever method, that is, using ^2, it can be orthogonalized at the same time, and then using the uniqueness of the arithmetic square root of the semi-positive definite matrix to prove that this method is not only more natural, but also simpler.

Jing Peng seemed bored and stayed in the study room all morning, leisurely looking through the various books that Jiang Shuiyuan placed on the bookshelf. When Jiang Shuiyuan finished the 13th question, he immediately reached out and took the answer sheet and read it carefully. Jiang Shuiyuan complained: "Teacher Jing, didn't you say it's easy to open the book? Why does it feel like it's like it's closed the book in the name of open book?"

"I just take a look at it casually, is it that you are in the way of searching for information?" Jing Peng calmly adjusted his sitting position.

"It doesn't stop me from searching for books and materials, but it affects my mood for exams!"

"After taking the exam, what else can I feel? Do the questions quickly!" Jing Peng was quiet for less than ten seconds and couldn't help but speak in surprise: "Hey, this solution is so exquisite, I never thought of it!"

Jiang Shuiyuan knew where he was talking: "I'm very clever? I thought of it only after suddenly getting inspiration."

Jing Peng nodded in praise: "No, not bad. But being able to think of this method can only be considered a little clever to finish the entire test paper on time, that is to truly learn advanced algebra."

What does it mean? It seems that the following question is not simple!

Facts have proved that Jing Peng was not a false threat. Jiang Shuiyuan felt more and more difficult to do as he went back. As soon as he started Question 15, he suddenly reached in front of him with one hand: "Food card!"

"Hmm?" Jiang Shuiyuan looked over with his hands and found that it was the average-looking low-level scientific research dog. Jing Peng had left at some point.

“What to eat for lunch?”

"Oh, it's already noon? Please give me a bibimbap and a box of milk, thank you."

"Okay." The low-level scientific research dog suddenly smiled with honey on his face and asked mysteriously, "In your class, are the class committee members and other things ranked according to their appearance? For example, the most handsome and beautiful person is the class monitor, and the second-ranked student committee member is the student committee member?"

"Yes, how do you know? I was still the team leader at that time!"

After eating the lunch sent by a gossiper enthusiastically, Jiang Shuiyuan continued to move forward slowly. If the last number of tests tests a person's patience and brain teasers, then this high-generation exam obviously focuses more on people's thinking ability and innovative application of knowledge. To put it another way, the last number of tests tests is like a knowledge competition, while this high-generation exam is like writing a paper.

At around nine o'clock in the evening, I finally made the last final question smoothly, proving the sufficient and necessary conditions for the interim sign of the three matrix rank inequality.

As we all know, matrix theory is an important part of higher algebra, and the rank of a matrix is ​​an important numerical parameter of a matrix. Before and after matrix operation, what kind of relationship will the rank of a matrix have? The textbook generally gives three important matrix rank inequality, namely:

rn&p;;rnrnrn

rn&p;;nrn,rn

rn&p;;rnrnrnn, where n is the number of columns of the matrix.
Chapter completed!