Chapter 007, I will just observe and not speak

The principal of Nangang No. 2 High School has fulfilled the commitment of "strengthening physical fitness and balanced development," ensuring that physical education classes will continue until the first semester of the third year of high school, whereas in some high schools, physical education classes are abolished in the second year of high school

Returning to the classroom of Class 2 in the second year of high school, Shen Qi began to tackle the exam questions set by Zhang Wanbang

System of equations → augmented matrix

Among them, the simplest is solid geometry, which can be understood as a three-dimensional version of high school plane analytic geometry

During the problem-solving process, Shen Qi encountered some obstacles. Performing elementary transformations on a system of linear equations is equivalent to applying row operations to its augmented matrix

In fact, Kailai is mistaken; this inference is incorrect. Both matrices must be indefinite for it to hold true. I only know the conclusion, but if you, Teacher Zhang, ask me to provide a proof, my abilities are limited and I am unable to do so

Then why are you studying it

I... I don't know!

Nangang No. 2 High School has a long history, established by patriotic revolutionaries during the Republic of China period. The principal strictly implements the fundamental principles of "strengthening physical fitness, balanced development, education for all, and treating everyone equally." The commitment to enhance students' physical well-being and promote positive energy education, regardless of their academic performance, is the principal's promise to society

Originally, the language class could earn 1 point of academic excellence, but Shen Qi did not pay attention during the lecture, so the contribution of this language class to the academic excellence points is 0

Problem 1: Prove the Cauchy-Schwarz inequality: XXXXXX (a mathematical expression that cannot be displayed on a mobile phone) and provide the conditions under which equality holds

"Oh, Shen Qi, what book are you reading?" Chen Xiaoting, the classmate sitting next to him, noticed that Shen Qi was reading a strange book

"Alright, Teacher Zhang, see you tomorrow." Shen Qi bid farewell to Zhang Wanbang and left the teacher's office

Shen Qi gritted his teeth and forced himself to solve the problem, a process that was extremely painful. However, by persevering through this hurdle, he reaped the rewards of a doubled sense of accomplishment

As a class teacher, Zhang Wanbang is puzzled; is this still the same underachiever Shen Qi, who was obsessed with games and lacked academic skills?

Shen Qi quickly completed the proof and wrote out the proof process on a blank sheet of paper

First-year students in the Department of Mathematics have three required foundational courses: Advanced Algebra, Mathematical Analysis, and Analytical Geometry

"My eyes are so blurry..." Shen Qi switched back and forth between the linear equations and their corresponding augmented matrices, to the point where he began to doubt his life.

After the conclusion of the ninth set of radio calisthenics, Shen Qi remained on the playground, as the next class was physical education

Here, A and B are not necessarily symmetric, but they comply with the condition that the absolute value of A + λB is not identically zero

I ... ... I cannot understand ... ...

"Oh, 2 points for academic excellence." Shen Qi can no longer earn academic excellence points by solving high school math problems, but he can earn them by tackling university-level math problems

It seems quite profound, but it has nothing to do with me. I only scored 85 points on this math test, and I truly regret being placed in the science class. Chen Xiaoting's grades in mathematics, physics, chemistry, and biology are all quite average; she relies on her scores in Chinese and English to pull up her total score, barely maintaining a ranking in the middle tier.

System: "Host successfully solved the problem, rewarded with 2 points of academic excellence credit"

Indeed, its inverse theorem originates from Sylvester and was proven by Weierstrass. If I remember correctly, our linear algebra textbooks from that era contained just one sentence about this inverse theorem. Do you know what that sentence is?

Then the third row of the augmented matrix undergoes a transformation

Let's go and do some exercises. Shen Qi stood up and called Chen Xiaoting to go downstairs together

As a mathematics teacher, Zhang Wanbang is delighted because his students are discussing Cayley and Weierstrass with him

The host successfully solved the problem, earning 5 points of academic excellence credit

Thus, the augmented matrix must also be transformed into ... ...

As for the subsequent branches such as real analysis, functional analysis, topology, partial differential equations, differential geometry, and algebraic geometry, those who have the capability should delve into them, while those who lack the ability will fail the course

The advanced generation offers a form of abstract thinking logic; to say it is useful, indeed it is very useful, but to say it is of no value, it is utterly worthless. In the face of the immensely vast realm of mathematics, Shen Qi is still an apprentice. What the ultimate meaning of mathematics is, Shen Qi cannot articulate clearly; he explores it in his own way of understanding.

Shen Qixin was wholly focused on solving Zhang Wanbang's math problem; he did not pay attention in Chinese class. After all, human energy is limited, making it difficult to multitask.

Eliminate the x1 term in the first equation

"Advanced Algebra." Shen Qi closed the book "Advanced Algebra" and pushed it onto Chen Xiaoting's desk

This discussion lasted for approximately 10 minutes, primarily with Zhang Wanbang asking questions and Shen Qi providing answers

"I will just take a look, without saying anything." Chen Xiaoting symbolically flipped through a few pages and then returned the book "Advanced Algebra" to Shen Qi: "Is this book 'Advanced Algebra' really your secret to achieving a perfect score?"

Meanwhile, the language teacher entered the classroom; this period is a language class

Chen Xiaoting was stunned when she saw the cover: "Advanced Algebra, Full-time University Mathematics Textbook, this... is this a textbook for university students?"

"Kailai or Weierstrass, matrix algebra or various determinants, it is difficult to explain clearly in just a few words." Zhang Wanbang casually took out a sheet of A4 white paper, wrote down a few lines of mathematical symbols, and then handed the paper to Shen Qi: "Do as much as you can, come to my office tomorrow at this time."

This inverse theorem states that if the elementary factors of the determinants of A + λB and A ' + λB ' are consistent, then it is possible to find a pair of linear transformations that simultaneously transform A into A ' and B into B ' How do you, Shen Qi, understand this inverse theorem?

Correct. Shen Qidian nodded

At this moment, the school broadcast began to play the athletes' march, signaling the start of the break exercise period

Actually, it's nothing serious, you should go and exercise. Chen Xiaoting's face turned red; how could a girl feel comfortable telling you boys about her needs

Oh, Aunt Flo has arrived, so indeed, exercising is not advisable; jumping around could lead to complications. Shen Qi has heard various rumors about Aunt Flo, and today's high school students understand everything: "Then you should rest well, and if you need anything, just let me know."

Chen Xiaoting smiled awkwardly while holding her lower abdomen: "I am not feeling well, so I will not be exercising today"

"Solving two math problems is equivalent to attending seven Chinese language classes. Doing 200 math problems for college students... never mind, I will take a break first." During the break, Shen Qi took out "Advanced Algebra" from his backpack to revisit the specialized courses of mathematics students.

Advanced mathematics is indeed too abstract for high school students, but I am pleased that you, Shen Qi, have been able to self-study to this level

For most boys, physical education classes are the happiest moments, a rare respite from the burdens of heavy academic work

It is not the case that this book provides direct assistance in improving high school mathematics scores

The second problem posed by Zhang Wanbang is to solve a system of linear equations, which requires a comprehensive application of the Gaussian elimination method and the properties of augmented matrices, thus increasing the level of difficulty

In fact, it doesn't matter much, because Shen Qi solved two math problems in the language class, earning 7 scholar points

After spending an entire language class and burning countless brain cells, Shen Qi finally solved the second problem posed by Zhang Wanbang

This question is not too difficult, it is an introductory proof problem from "Advanced Algebra", focusing on the concept of inner product spaces

In terms of abstraction, the most difficult subject is advanced algebra. In terms of computational complexity, the most challenging subject is differential calculus

Shen Qi took the white paper and found that it contained five math problems, indicating that Teacher Zhang intended to further test him

We generally consider Cayley to be the founder of matrix theory. Cayley has a theorem stating that the product of two matrices can be zero without either of them being zero, as long as one of them is singular. Shen Qi, do you think this theorem is correct?

Augmented matrix → system of equations

Add 4 times the second equation to the third equation to eliminate the x2 term in the third equation, resulting in a stair-step form of the system of equations

"Weierstrass was the first to obtain the standard form of the bundle A + λB. Shen Qi, how do you understand the standard form of this bundle?"

Within a few minutes, over a thousand students gathered on the school playground, lined up according to their classes