Science Building, lecture hall.
Professor Liu Ruochuan leaned on the lectern with both hands, a smile on his face as he watched the students, whose CPUs were practically smoking from the mental effort.
To be honest, he hadn’t expected any of these newly enrolled freshmen to be able to solve the problem he’d given them.
’Testing your level with a question from the graduate school entrance exam isn’t going too far, is it...’
Professor Liu’s heart itched with anticipation.
’This *is* Yuanpei College, after all. What if some monster can actually solve it? As the Deputy Dean of the School of Mathematics, poaching a few promising talents shouldn’t be a problem...’
He watched as not a single person below seemed to have a clue, their brows all deeply furrowed.
Professor Liu smiled and decided to add a little fuel to the fire.
’Teaching is all about having fun...’
"This problem, you see, isn’t actually beyond the scope of the curriculum."
"As long as you paid close attention and fully grasped the concepts of subspace intersection and sum that I taught in this class, you can definitely solve it."
"Therefore, I want to encourage your courageous attitude toward this challenge."
"Whoever can come up here and solve it, I’ll give you full marks for participation for the rest of the course!"
Down below, the heads of these chosen prodigies were buzzing.
’"If we fully grasped it, we could solve it? Is Professor Liu messing with us?"’
’"Full participation credit? He’s offering a pie we can’t even take a bite of!"’
But just as Professor Liu finished speaking, a figure in the front row stood up.
Liu Qiang and Chen Nan, sitting beside him, were stunned, their jaws slightly agape.
Wang Hao, the number one math student of Dorm 404, also looked up in astonishment.
Li Dong spoke.
"Teacher Liu, I’d like to try."
It wasn’t that he wanted to show off. The main reason was that his passion was physics, and if his physics and math classes conflicted in the future, he might not be able to get full participation credit in math.
Professor Liu hadn’t quite seen what had shot up right under his nose and paused for a moment.
He had a faint impression of Li Dong and knew his surname was Li.
It wasn’t because Li Dong was the top science scholar from Sichuan and Chongqing.
After all, at Yuanpei, you could throw a brick and hit three top scholars.
It was because on the second day of the semester, this student had asked his counselor for a leave of absence, saying he was going to Zhejiang University to advance a biology project.
To be able to travel to another province to independently advance a research project as a freshman... This student’s ability was formidable....
So, he also felt a sliver of anticipation.
"Alright, come on up and give it a try."
Li Dong walked up to the lectern and, without a second thought, began writing on the blackboard.
He knew where the trap in this problem lay.
Many people’s first reaction to this problem would be to extend the bases of V1 and V2 separately, but this would lead to the constructed W being unable to satisfy both direct sum conditions simultaneously.
The real entry point was to first deal with their intersection space.
Step 1: Handle the intersection and extend the basis.
Li Dong wrote down the core idea: First, take a basis for V1∩V2, then use the Basis Extension Theorem for Subspaces to extend it into a basis for V1 and a basis for V2, respectively.
Professor Liu stood to the side, nodding to himself.
’His fundamentals are solid, and his intuition is sharp. He avoided the biggest trap with the very first step.’
Step 2: Prove linear independence.
This was the toughest bone to gnaw on in the entire problem.
Li Dong didn’t get bogged down in tedious calculations but went straight to the heart of the matter. He set up an equation where a linear combination of the vectors from set b and set c equaled zero. With a clever transposition of terms, the left side of the equation belonged entirely to V1, while the right side belonged entirely to V2. This meant both sides must naturally belong to the intersection V1∩V2.
Next, by substituting the basis of the intersection from the first step and leveraging the linear independence of the known basis vectors, he deduced in just a few steps that all the combination coefficients must be zero.
He had cleanly proven that the combination of the vectors from set b and set c was still linearly independent!
Step 3: Construct the complementary subspace W.
Since the combined vectors from the beta and gamma sets were still linearly independent, he could use the Basis Extension Theorem once more to extend them, along with the basis of the intersection space, into a basis for the entire space V!
On the blackboard, Li Dong circled the vectors of the beta and gamma sets, added a plus sign between each pair of β_i and γ_i, and then circled the brand-new set of eta vectors that were added during the final basis extension.
"Let W be the subspace generated by this set of combined vectors β_i + γ_i and the newly added eta vectors."
Following that were a few lines of concise dimensional calculations.
For both V1 and V2, their dimension plus the dimension of W was exactly equal to the total dimension n, and their intersection with W was the zero space.
Therefore, V = V1⊕W = V2⊕W.
A subspace W satisfying the conditions exists. Q.E.D.!
RINGGG.
The dismissal bell rang at almost the exact moment Li Dong wrote the final character.
Professor Liu Ruochuan stood in place, carefully reviewing the proof on the blackboard, then turned his head to look at Li Dong.
"Very nice."
Those two words, for a new undergraduate, were a tremendous affirmation.
At this moment, the top-tier prodigies in the audience were all dumbfounded.
Even though the proof on the board contained some advanced concepts they hadn’t fully learned yet, they could already see the method behind the madness, thanks to their own freakish powers of logical deduction.
Wang Hao sat in his seat, looked at the blackboard, and finally let out a sigh of resignation...
Hearing the bell, Li Dong casually put down the chalk.
"Li Dong, you really do have a thorough understanding."
Professor Liu asked with a smile.
"So, in your opinion, what was the difficult part of this problem?"
Professor Liu’s intention was to use Li Dong’s explanation to point out the problem’s traps and help clarify the thought process for the students who hadn’t quite caught on yet.
However, Li Dong shook his head and said with great sincerity, "This problem... didn’t really have any difficult parts."
Professor Liu was taken aback.
"Huh?" The students below all rolled their eyes in unison.
’Damn,’ they thought, ’he’s not even pretending anymore. Just laying his cards on the table, huh?’
Li Dong genuinely didn’t find the problem difficult, but to answer Professor Liu, he thought for a moment and added something.
"That problem just now really wasn’t hard."
"But if the question went a little deeper and asked whether the subspace W that satisfies the above conditions is unique... now *that* might be a little interesting."
As soon as Professor Liu heard this, being an academic heavyweight, he instantly understood what Li Dong meant.
He asked back with great interest, "And what do you think? Is it unique?"
The students below also began to follow Li Dong’s line of thought.
Wang Hao quickly calculated in his head and muttered under his breath.
"Since it’s constructed by extending the basis, it should be unique, right...?"
The few other students who could keep up with Li Dong’s thinking had the same idea.
Li Dong shook his head. "It’s not unique."
The students below who thought it was unique were instantly confused.
Li Dong casually picked up the chalk again and, in a blank space on the board, dashed out a counterexample in just a few strokes: Let V = the two-dimensional real vector space, V1 = L((1,0)), and V2 = L((0,1)). Then W1 = L((1,1)) and W2 = L((1,2)) both satisfy the direct sum condition, directly disproving uniqueness!
Professor Liu’s eyes were filled with admiration.
"Mm, not bad at all," he said, then muttered to himself.
’Next class, when I teach the other sections, I’ll add this second part to the question...’
The students from the other classes: ??? How rude!
"Alright, class dismissed." Professor Liu clapped the chalk dust from his hands.
"When you have time later, take a good look at the proof on the blackboard and think through the logic."
Then he turned his head toward Li Dong. "Student, what’s your full name?"
"My name is Li Dong."
Professor Liu smiled and nodded.
"Li Dong, is it? When it’s time to choose your major specialization in your sophomore year, you should give serious consideration to our School of Mathematical Sciences. You could work on a research project with me then."
"Thank you, Teacher Liu. I’ll consider it." Li Dong expressed his thanks politely and then hurried down from the lectern.
As he passed the front row, he called out to his three roommates.
"I’m heading back first! Don’t worry about getting food for me!"
Li Dong was in a hurry to get back, organize his data, and write his paper.
Although the food at Nongyuan Cafeteria was quite delicious, squeezing into the cafeteria at this time would be too much of a waste.
He planned to just go to the basement of Building 35 and grab a quick bite to eat.
The facilities at Yuanpei’s Building 35 were superb. It not only had light meals and coffee, but also a gym, a basement, and conference rooms—it was fully equipped.
Only after Li Dong had disappeared from the classroom doorway did Chen Nan and Liu Qiang snap back to reality from their stupor over the counterexample on the blackboard.
The two of them turned their heads at the same time, looking at Wang Hao with complicated expressions.
"Haozi..."
"I remember at the beginning of the semester, Brother Dong told us his favorite subject was physics, right?"
Wang Hao stared at the ceiling with a look of utter despair.
"Yeah..."

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