I'm Just Looking, Not Talking

As a homeroom teacher, Zhang Wanbang was puzzled. Was this still the same Shen Qi who was obsessed with games and utterly useless?

As a math teacher, Zhang Wanbang was delighted because his student was discussing Cayley and Weierstrass with him.

This discussion lasted about ten minutes, with Zhang Wanbang mostly asking questions and Shen Qi answering.

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

"Actually, Cayley was wrong. This corollary is incorrect; both matrices must be indefinite. I only know the conclusion. Teacher Zhang, if you want me to provide a proof, my skill level is limited, and I can't do it."

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

"A and B here are not necessarily symmetric, but they must satisfy the condition that the absolute value of A + λB is not identically zero."

"That's right. The converse theorem comes from Sylvester and was proven by Weierstrass. If I remember correctly, our Advanced Algebra textbook from our era only wrote one sentence about this converse theorem. Do you know that sentence?"

"I... I don't know!"

"This converse theorem states that if the determinant of A + λB is elementarily equivalent to the determinant of A' + λB', then a pair of linear transformations can be found that transform A to A' and B to B'. Shen Qi, how do you understand this converse theorem?"

"I... I can't understand it..."

"Advanced Algebra is indeed too abstract for high school students, but I am delighted that Shen Qi, you can teach yourself to this level."

"Cayley or Weierstrass, matrix algebra or various determinants, it's hard to explain clearly in a few words." Zhang Wanbang casually took out a sheet of A4 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 at this time tomorrow."

Shen Qi took the paper and found five math problems written on it. It seemed Teacher Zhang wanted to test him further.

"Okay, see you tomorrow, Teacher Zhang." Shen Qi said goodbye to Zhang Wanbang and left the teacher's office.

Returning to the classroom of Class 2, Grade 2, Shen Qi began to tackle the problems Zhang Wanbang had assigned.

The first problem was to prove the Cauchy-Schwarz inequality: [mathematical expression that cannot be displayed on a mobile phone], and to give the conditions for equality.

This problem wasn't too difficult; it was an introductory proof problem from "Advanced Algebra" that tested the concept of inner product spaces.

Shen Qi quickly completed the proof and wrote down the steps on the paper.

System: "Host successfully solved the problem, reward 2 scholar points."

"Oh, 2 scholar points." Shen Qi could no longer get scholar points for solving high school math problems, but he could get them for solving university-level math problems.

At the same time, the Chinese teacher walked into the classroom. It was time for Chinese class.

Shen Qi was focused on solving Zhang Wanbang's math problems and didn't pay attention to the Chinese lesson. A person's energy is limited, and it's hard to do two things at once.

Zhang Wanbang's second problem was to solve a system of linear equations, which required the comprehensive application of Gaussian elimination and the properties of augmented matrices. The difficulty had increased.

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

System of equations → Augmented matrix

Augmented matrix → System of equations

Eliminate the x1 term in the first equation.

Then the third row of the augmented matrix undergoes a transformation.

Add 4 times the second equation to the third equation to eliminate the x2 term in the third equation, resulting in an echelon form system of equations.

Then the augmented matrix must also be transformed into...

"My eyes are so blurry..." Shen Qi switched back and forth between the system of linear equations and its corresponding augmented matrix, to the point of doubting his life.

Shen Qi gritted his teeth and forced himself to solve the problem. The process was extremely painful, but by persevering, he overcame this hurdle and reaped double the satisfaction.

After spending an entire Chinese class period, burning countless brain cells, Shen Qi finally solved Zhang Wanbang's second problem.

"Host successfully solved the problem, reward 5 scholar points."

Originally, he could have earned 1 scholar point for the Chinese class, but since Shen Qi hadn't listened attentively, this Chinese class contributed 0 scholar points.

It didn't really matter, though, because Shen Qi had solved two math problems during Chinese class, earning a total of 7 scholar points.

"Solving two math problems is equivalent to attending seven Chinese classes. Solving 200 university math problems... Forget it, I'll rest for a while." During the break, Shen Qi took "Advanced Algebra" out of his backpack and revisited the professional courses for mathematics majors.

First-year mathematics students had three compulsory foundational courses: "Advanced Algebra," "Mathematical Analysis," and "Analytic Geometry."

Among them, Analytic Geometry was relatively the simplest, understandable as a three-dimensional version of high school Plane Analytic Geometry.

In terms of abstraction, Advanced Algebra was the most difficult. In terms of calculation, Mathematical Analysis was the most difficult.

As for subsequent branches like Real Analysis, Functional Analysis, Topology, Partial Differential Equations, Differential Geometry, and Algebraic Geometry, whoever had the ability could delve into them, and whoever lacked the skill would fail.

"Huh, Shen Qi, what book are you reading?" His deskmate Chen Xiaoting noticed Shen Qi reading a strange book.

"Advanced Algebra." Shen Qi closed "Advanced Algebra" and pushed it across to Chen Xiaoting's desk.

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

"Yes." Shen Qi nodded.

"I'll just take a look, no talking." Chen Xiaoting flipped through a few pages symbolically and then returned "Advanced Algebra" to Shen Qi. "Is this 'Advanced Algebra' your secret to getting perfect scores?"

"No, this book doesn't directly help improve high school math scores."

"Then why are you studying it?"

"Advanced Algebra provides an abstract logical thinking process. It's very useful if you say it's useful, and worthless if you say it's useless." In the face of the immensely vast field of mathematics, Shen Qi was still an apprentice. He couldn't quite explain the ultimate meaning of mathematics, and he explored it in his own way.

"That sounds really profound, but it has nothing to do with me. I only got 85 on my math test this time, I really regret being put in the science class." Chen Xiaoting's grades in math, physics, chemistry, and biology were all average. She relied on Chinese and English to pull her score up, barely maintaining a middle-of-the-pack ranking overall.

Just then, the school broadcast began playing the march of the athletes; it was time for the break exercise.

"Let's go, time for exercises," Shen Qi stood up, calling to Chen Xiaoting to head downstairs together.

Chen Xiaoting clutched her lower abdomen and smiled awkwardly, "I'm not feeling well, I'm not doing exercises today."

Oh, her period is here. Then she really can't do exercises; jumping around could cause problems. Shen Qi had heard all sorts of rumors about periods. Today's high school students knew everything. "Then you get some rest. Tell me if you need anything."

"Actually, it's not a big deal. You should go do your exercises," Chen Xiaoting's face flushed. How could she tell the boys about a girl's needs?

Within a few minutes, thousands of students had gathered on the school's large sports field, lined up by class.

Nangang Second High School had a long history, founded during the Republic of China era by patriotic revolutionary figures. The principal strictly implemented the basic policy of "strengthening the body, balanced development, educating all without discrimination, and treating everyone equally." There were no fast or slow classes. Strengthening students' physique and educating them with positive energy were the principal's promises to society.

After the ninth set of broadcast exercises concluded, Shen Qi remained on the field; physical education class was next.

For most boys, physical education class was the happiest time, a rare respite from their heavy academic workload.

The principal of Nangang Second High School fulfilled his promise of "strengthening the body, balanced development." Physical education classes would be retained all the way until the first semester of senior year. In contrast, at some high schools, physical education classes were abolished in the second year of high school.