Advanced Algebra?

The room was so quiet that only the soft scratch of pen on paper remained.

The sunlight outside was a little glaring. Lu Yao pulled the curtains halfway shut, then turned and wrote two words in her notebook—[Advanced Algebra].

"Junior, since you want to understand the structure of a university mathematics department, we have to start with this course."

Since Lu Yao had decided to explain it, she brought out every bit of her professional attitude.

"Advanced Algebra and Mathematical Analysis are the two great mountains facing first-year students."

"The calculus you want to study is really only one part of Mathematical Analysis, while Advanced Algebra focuses on transformations of spaces."

"For example, the definition of linear spaces, changes of basis and coordinate transformations—those are all fundamentals of Advanced Algebra."

As Lu Yao spoke, she listed several core concepts on the paper.

"And there's actually a difficult point here: extending a maximal linearly independent set. Understand?"

Li Dong looked at the concept, then connected it with the materials he had come across while studying New Lectures on Mathematical Analysis on his own, and instinctively nodded.

"I understand. It's the rank of a vector set."

Lu Yao paused. It was the same thing, but how could a high school student react that quickly?

"Fine, then let's continue."

"Next is the matrix representation of linear transformations. That's the key topic for the second semester of freshman year."

"This involves eigenvalues and eigenvectors, as well as diagonalizing matrices..."

"I understand." Li Dong nodded again.

"The roots of the characteristic polynomial."

Lu Yao: "..."

She took a deep breath, and the hand holding her pen unconsciously tightened.

"Fine. Then let's go deeper. What if it's a linear space over the field of complex numbers, involving Jordan Canonical Form?"

"I understand. Elementary divisors and invariant factors."

"What about orthogonal transformations in Euclidean Space?"

"I understand. Transformations that preserve the inner product."

"Bilinear functions?"

"I understand."

Half an hour later.

Lu Yao felt like she was about to lose her mind.

She had originally wanted to show this junior just how vast mathematics was.

But what happened?

She felt like a tour guide who had just pointed at a mountain and said, "This mountain is hard to climb," only for Li Dong to already be standing at the summit yelling, "Next one!"

And every one of Li Dong's "I understand"s felt like a point-blank ultimate move to her face.

That's it? Just that?

"Y-you really understood all of it?"

Lu Yao was utterly unwilling to believe it.

"Mm. I've roughly sorted out the logic behind it all."

Li Dong looked at her with admiration.

"Senior, you explained it so well. It's much clearer than those roundabout definitions in the books."

When he had studied on his own before, his knowledge points had actually been a little scattered. If his Logic attribute hadn't increased by 0.1, he wouldn't have been able to connect many of them.

What Lu Yao was doing now was giving him a thread and stringing all those pieces of knowledge together.

So Li Dong was telling the truth.

But to Lu Yao, it sounded a little like humblebragging.

"Fine, then I'll see whether you really understand or you're just pretending!"

Lu Yao was getting a little angry.

She no longer cared about progressing step by step. She skipped straight past the foundational courses of freshman and sophomore years and set her sights on a junior-year specialized course—Real Analysis.

As the saying went, Real Analysis took ten tries to learn, so its difficulty was obvious.

"Then listen carefully."

Lu Yao became adorably fierce.

"Tell me, what is the Cantor Ternary Set? What is its measure?"

Li Dong answered without hesitation.

"Take the closed interval [0,1], remove the middle third, and repeat the process. Its measure is 0, but its cardinality is c, the cardinality of the continuum, so it's an uncountable set."

Lu Yao's eyes widened.

"Then... tell me the difference between the Lebesgue Integral and the Riemann Integral."

"With Riemann integration, we slice vertically. With Lebesgue integration, we slice horizontally, so Lebesgue integration can handle more types of functions, such as the Dirichlet Function."

Li Dong still spoke with complete certainty.

"Gasp..."

Lu Yao sucked in a breath of Yang Shengguo.

The same thought Old Yang had once had surfaced in her mind.

Was this kid a monster?

It had taken her more than half a semester to understand those concepts.

Li Dong looked at Lu Yao's heaving chest and felt utterly baffled.

What was wrong with this senior?

She had clearly been rather gentle just now. Why had she suddenly become so... fierce?

Lu Yao was already getting carried away.

"Fine! You've memorized the theory pretty well, huh? Then let's put it into practice!"

She directly pulled her own review materials, Typical Problems and Methods in Mathematical Analysis, from her bag.

This book was famously difficult in the graduate entrance exam world.

"Come on! This problem!"

Lu Yao pointed at a proof problem concerning the convergence of a series.

Let a_n > 0, and let the series ∑ a_n diverge. Prove that the series ∑(a_n / S_n^2) converges, where S_n is the sum of the first n terms.

After reading the problem, Li Dong was slightly stunned as well.

If he had encountered this problem before today, he might have gotten stuck for a moment.

But now...

Wasn't this just the discrete form of the integral test?

"What? No idea, huh?"

Seeing Li Dong remain silent, Lu Yao finally felt a little better.

Then, just as she was about to properly give this cocky junior a taste of a senior's power—

"This problem requires an extension of the Cauchy Convergence Criterion, and also..."

"I've got it."

Li Dong picked up his pen and began writing directly on the scratch paper.

"This problem actually doesn't need to be so complicated."

"Look, a_n = S_n - S_{n-1}."

"So the general term a_n / S_n^2 can be bounded."

Li Dong muttered as he wrote.

"a_n / S_n^2 < a_n /(S_n * S_{n-1})=(S_n - S_{n-1})/(S_n * S_{n-1})= 1/S_{n-1}- 1/S_n"

After writing that line, Li Dong looked up at Lu Yao.

"Now it telescopes."

"When you sum the first N terms, everything in the middle cancels out, leaving only 1/S_0 - 1/S_N."

"Since ∑ a_n diverges, S_n tends to infinity, and 1/S_n tends to 0."

"Therefore, the series converges."

Li Dong put down his pen.

The air in the room suddenly went quiet.

Lu Yao stared blankly at the scratch paper.

She had spent half an hour on it last night, trying all kinds of complicated ratio tests, and yet...

Seeing Lu Yao remain silent for so long, Li Dong thought he had made a mistake and asked uneasily.

"Senior? Is it wrong? I thought the structure looked a lot like the telescoping sums we learned in tenth grade, so I gave it a try..."

Lu Yao came back to her senses and began muttering to herself.

"Bounding it... telescoping... that works too?"

"It really does converge... This answer is correct..."

She raised her head and stared at Li Dong with resentful eyes.

Li Dong was left at a loss by her stare, thinking to himself.

Why is Senior looking at me like that?

Why does that look feel kind of like Mi Xia's? Like she wants to eat me.

Are pretty girls all this strange?

Finally, Lu Yao stammered out, "Y-you... How did you think of splitting the denominator into a product of two terms and then bounding it? Th-that's a highly technical move!"

Li Dong scratched his head.

"Didn't you just say we needed to learn how to 'construct' things? I was thinking about how to cancel out the squared term, so I tried it."

Lu Yao felt like she had taken ten thousand points of damage.

Over the next stretch of time, the whole atmosphere gradually went off the rails.

Lu Yao: "What about this one? What if it's an alternating series?"

Li Dong: "Oh, just use the Leibniz Test. Look, it's monotonically decreasing and tends to zero..."

Lu Yao: "What about changing the order of integration in this double integral?"

Li Dong: "Just draw it out. Integrate with respect to y first, then x..."

Lu Yao nodded frantically beside him, her pen constantly taking notes. She had completely forgotten that she was supposed to be the tutor being paid.

"Grrr..."

An ill-timed sound broke the academic atmosphere.

A flash of red crossed Lu Yao's face, and the pen in her hand stopped.

Li Dong checked the time and only then realized that it was already past one in the afternoon.

"Um... sorry, Senior. I lose track of time whenever I start solving problems."

Li Dong smiled apologetically.

"My mom isn't home, so I usually cook instant noodles for lunch. If you don't mind..."

Lu Yao glanced at Li Dong.

Her dignity as a third-year senior had already been shattered into pieces.

If she couldn't beat this junior at math, then she at least had to beat him at cooking.

"Why eat instant noodles? They're not nutritious."

Lu Yao stood up and rolled up her sleeves.

"I'll cook! Let you taste your senior's skills!"

This came as a pleasant surprise to Li Dong.

"You can cook?"

Li Dong looked at Lu Yao, a flash of admiration in his eyes.

"As expected, college students are different! Not only are you good at solving problems, you can cook too!"

"Once I get to college, will I be able to learn how to cook too?"

If some university president learned that Li Dong's longing for college came from thinking that "college teaches you how to cook," he would probably be so furious that he would elbow the Earth on the spot.

"Of course!"

Lu Yao confidently walked into the kitchen.

However, five minutes later.

The kitchen was filled with smoke. Lu Yao held a spatula in one hand and her phone in the other, its screen clearly playing "Homestyle Tomato Scrambled Eggs."

"Do I add the eggs first or the tomatoes first? Ah! The oil splashed out!"

Li Dong: "..."

He cautiously asked, "Senior Lu Yao, when you said you could cook... did you mean learning it on the spot by following a phone tutorial?"

With a bang, Lu Yao slammed the kitchen door shut.

"None of your business! Just wait outside!"

Half an hour later.

Two bowls of rice sat on the dining table, with a plate of strangely colored tomato scrambled eggs in the middle.

Li Dong picked up a blackened piece of egg with his chopsticks, hesitating slightly.

"Senior Lu Yao, you spent an hour in the kitchen... just to make this... charcoal-grilled tomato scrambled egg?"

Lu Yao glared at him, took a mouthful of rice herself, and said indistinctly.

"Eat it or don't!"

The two of them ate the meal in unusual silence.

After finishing, Lu Yao couldn't stay a moment longer.

The blow she had suffered today was simply too great.

And the approaches Li Dong had just explained to her—especially that "Partial-Fraction Scaling"—had left her mind in a mess. She needed to go back and properly digest and consolidate it.

"Um... I'll be going now. You get some proper rest too."

Lu Yao grabbed her bag and was about to make a run for it.

"Wait!"

Li Dong suddenly called after her.

Li Dong ran back into the bedroom and returned shortly afterward with two red banknotes.

"Lu Yao, my mom left this for you. She told me to give it directly to you."

Li Dong held out the money.

"Last time was one hour, and this time was about two hours. That's 150 yuan total. You even made me a meal, so I'll give you 200. Keep the change."

Li Dong was focused solely on completing the task his mother had assigned him, completely oblivious to how spectacular Lu Yao's expression had become.

Looking at the bright red bills, Lu Yao only felt her face burn with pain.

This 150 yuan... was he making a point of it?

He was clearly the one explaining problems to me today, wasn't he?

And now he's giving me money?

Lu Yao looked up at Li Dong and found his gaze incomparably clear, without the slightest trace of mockery.

"Ha..."

Lu Yao took a deep breath and pushed Li Dong's hand away.

"No need!"

"Keep the money and buy yourself some books!"

With that, she rushed out the door as though fleeing.

Before closing the door, she even turned back and shouted something.

"Sure!" Li Dong readily agreed.

Looking at the closed door, Li Dong was in an excellent mood.

For him, today's tutoring session had been an absolute gold mine.

The Mathematical Analysis and Advanced Algebra he had crammed down by force in the past had finally been thoroughly brought to life through Lu Yao's problem-by-problem tutoring.

"Sure enough, college students are different."

He recalled how Lu Yao had been angry but still explained everything earnestly, as well as that plate of tomato scrambled eggs that had been awful—truly awful.

The corners of Li Dong's lips curled up slightly.

"Although Lu Yao and Mi Xia both keep staring at me for no reason."

"I think I like Senior Lu Yao's type better."

"After all, she's a little older, knows more, and has a more complete system of knowledge..."

Monday was a hard day for most of the students at Jiangcheng No. 7 High School.

Because the monthly exams were coming.

In the classroom, the poorer students sighed and groaned, each looking as though they were heading to the execution ground.

But some people were exceptionally fired up.

Over the past month, Jiang Yizhou had hired a tutor, worked through countless sets of problems, and had even uninstalled Honor of Kings.

What was it all for?

It was to ruthlessly stomp that suddenly emerged Li Dong beneath his feet in this monthly exam!

Yet even when the bell for the first Chinese exam rang.

The window-side seat in the back row remained empty.

Li Dong hadn't come today.

The feeling of throwing a punch into cotton made Jiang Yizhou miserable.

"Hmph, avoiding my edge?"

Mi Xia in the front row also turned to glance at the empty seat.

A hint of disappointment flashed through her eyes.

"360 points? Is that for real?"

At this very moment.

In a computer room in the teaching building.

Li Dong sat before a computer, its screen displaying:

[The 16th "Huaxuan Cup" National Physics Innovation Competition (Preliminary Round)]

[Candidate: Li Dong]

[Exam Begins]