The room was so quiet that the only sound was the RUSTLE of a pen tip on paper.
The sunlight from outside was a bit harsh, so Lu Yao drew the curtains halfway. She then turned and wrote an abbreviation on her notepad—[Adv. Alg.].
"Since you want to understand the structure of a university math department’s curriculum, we have to start with this course."
Since Lu Yao had decided to explain, she adopted an utterly professional attitude.
"Advanced Algebra and Mathematical Analysis—those are the two great mountains for us first-year students."
"The Calculus you want to study is actually just a part of Mathematical Analysis, while Advanced Algebra deals with transformations of space."
"For example, the definition of linear space, basis transformation, and coordinate transformation... these are the fundamentals of Advanced Algebra."
As she spoke, Lu Yao jotted down a few core concepts on the paper.
"And within this, there’s a difficult part: the extension of a maximal linearly independent set. Understand?"
Li Dong looked at the concept. Recalling the supplementary materials he’d seen while self-studying *New Introduction to Mathematical Analysis*, he nodded subconsciously.
"I get it. That’s just the rank of a vector set."
Lu Yao froze for a moment. ’They’re the same thing, but how can a high school student react so quickly?’
"Fine. Let’s continue."
"Next is the matrix representation of linear transformations. This is a key topic in the second semester of the first year."
"This involves eigenvalues and eigenvectors, as well as matrix diagonalization..."
"Got it," Li Dong nodded again.
"The roots of the characteristic polynomial."
Lu Yao: "..."
She took a deep breath, her hand unconsciously tightening its grip on the pen.
"Okay, let’s go deeper. What about linear spaces over a complex field, involving the Jordan normal form?"
"I get it. Elementary divisors and invariant factors."
"What about orthogonal transformations in Euclidean space?"
"Understood. A transformation that preserves the inner product."
"Bilinear function?"
"Got it."
「Half an hour later.」
Lu Yao felt like she was about to go crazy.
She had originally wanted to show this junior just how vast mathematics was.
And the result?
She felt like a tour guide who had just pointed to a mountain and said, "This one is hard to climb," only for Li Dong to already be standing on the summit yelling, "Next!"
And every time Li Dong said "got it," it was like he was unleashing an ultimate ability right in her face.
’This? That’s it?’
"You... you really understood all of that?"
Deep down, Lu Yao refused to believe it.
"Yeah, I’ve pretty much straightened out the overall Logic."
Li Dong looked at her with a face full of admiration.
"You explain things so well, Senior. It’s much clearer than all the convoluted definitions in the books."
He had been self-studying before, and the concepts were actually a bit of a mess in his head. If it weren’t for his +0.1 Logic attribute, he wouldn’t have been able to connect many of them.
What Lu Yao had just done was give him the thread to string all those concepts together.
So, Li Dong was telling the truth.
But to Lu Yao’s ears, it sounded a lot like humblebragging.
"Fine! Then I’d like to see if you really understand or are just faking it!"
Lu Yao was getting a little angry.
She abandoned any thought of a gradual progression, skipping right over the first and second-year fundamental courses and aiming straight for a third-year specialized course—Real-Valued Functions.
As the saying goes, "You have to study Real-Valued Functions ten times over," so one can only imagine its difficulty.
"Alright, listen up."
Lu Yao was now getting cutesy-fierce.
"Tell me, what is the Cantor ternary set? And what is its measure?"
Li Dong blurted out the answer.
"You take the closed interval [0, 1], remove the middle third, and repeat the process indefinitely. Its measure is 0, but its cardinality is c (the cardinality of the continuum), so it’s an uncountable set."
Lu Yao’s eyes went wide.
"Then... tell me the difference between the Lebesgue integral and the Riemann integral."
"With Riemann, we slice vertically. With Lebesgue, we slice horizontally. That’s why Lebesgue can handle more types of functions, like the Dirichlet function."
Li Dong said with complete confidence.
"Hiss..."
Lu Yao gasped.
A thought, the same one Old Yang once had, popped into her head.
’This kid... is he a monster?’
She had spent half a semester just to properly grasp these concepts!
Li Dong watched Lu Yao’s chest heave up and down, feeling a little baffled.
’What’s wrong with this senior?’
’She was so gentle just a moment ago. Why did she suddenly become so... fierce?’
Lu Yao was getting worked up now.
"Fine! So you’ve memorized the theory, have you? Then let’s move on to a practical application!"
She pulled her review materials right out of her bag: *Typical Problems and Methods in Mathematical Analysis*.
This book was notorious in the grad school exam prep world as a tough nut to crack.
"Here! This problem!"
Lu Yao said, pointing to a proof problem on the convergence of a series.
[Let a_n > 0, and the series ∑a_n diverge. Prove that the series ∑(a_n / S_n^2) converges, where S_n is the partial sum of the first n terms.]
After reading the problem, Li Dong paused for a moment.
If he had seen this problem before today, he might have gotten a little stuck.
But now...
’Isn’t this just using the discrete form of the integral test?’
"What’s the matter? No ideas?"
Seeing Li Dong remain silent, Lu Yao finally felt a little better.
Just as she was preparing to give this cocky junior a dose of shock and awe from his senior...
"This problem requires the generalization of the Cauchy convergence principle, and also..."
"I’ve got it."
Li Dong picked up his pen and started writing directly on the scratch paper.
"This problem doesn’t need to be that complicated."
"Look, a_n = S_n - S_{n-1}."
"So we can bound the general term a_n / S_n^2."
Li Dong muttered as he wrote.
"a_n / S_n^2
After writing this line, Li Dong looked up at Lu Yao.
"And now it’s a telescoping series."

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