欧美a欧美a

Today, there was an additional beneficiary—Chen Yiwen.

For Qiao Ze, this wasn\'t considered a burden.

Explaining these simple fundamental concepts was just the right way to relax his brain.

...

"Ask," As the three of them got ready, Chen Yiwen heard Qiao Ze say these simple two words to Su Mucheng, then unconsciously looked towards the two of them.

Su Mucheng, who hadn\'t seemed to be paying much attention in the afternoon, now became serious and pushed her notebook to the middle between her and Qiao Ze. Then she leaned her entire head in closer: "Qiao Ze, I didn\'t quite understand the part about matrix dimensionality reduction in linear algebra today, it seems a bit different from what\'s in our textbook?"

Experience tales at m v|l e\'-NovelBin.net

"Show me the textbook," Qiao Ze said.

Su Mucheng found the textbook from her bag and handed it over.

Qiao Ze glanced at the table of contents and flipped through a few pages, then said, "Got it."

He then asked, "What\'s the essence of a matrix?"

Su Mucheng replied, "A function?"

Qiao Ze emphasized, "More accurately, it\'s a function for operating on vectors."

Su Mucheng nodded.

"What does it do?"

"Transformation and scaling, but what does this have to do with dimensionality reduction? A two-dimensional vector still is a two-dimensional vector after rotation and stretching, right?"

"That\'s the perspective of the textbook. The matrix only acts on a single vector separately, this way of learning is skewed toward teaching you how to solve problems in the simplest way. But in fact, the more advanced view in the mathematical community now is that a matrix can be a transformation that acts on the entire space.

During the transformation, the single vector you observe will deform with the deformation of the space."

"Huh?" Su Mucheng listened even more intently.

Qiao Ze picked up a piece of paper and a pen and began to explain.

"Once you understand this concept, look at this special matrix. We use this matrix to operate on this two-dimensional space, which can also be understood as operating on all the points within that two-dimensional space. You see, after the operation, the overall area of the space equals zero, that is to say, the space has been compressed into a line."

"Isn\'t this just like a two-dimensional foil?" Su Mucheng said in surprise.

Qiao Ze frowned and asked, "What\'s a two-dimensional foil?"

"Ah? Haven\'t you read \'Three-Body\'?"

As Su Mucheng chattered on to explain the fantastical weapon known as the two-dimensional foil, Qiao Ze shook his head.

"While it\'s true that dimensionality reduction matrices do have applications in physics, they are mainly in the quantum dimension and do not affect the macroscopic.

Additionally, the correct mathematical expression is that under the operation of a dimensionality reduction matrix, a subspace of a two-dimensional space is mapped into zero-degree space, which is the kernel as explained in the textbook."

Su Mucheng nodded as if suddenly enlightened.

"The principle is the same, whether it\'s reducing from two dimensions to one, from three to two, or from four to one. The issue now is how to determine a dimensionality reduction matrix. There are two methods, the simpler one is using determinants; the other is by comparing the rank with the dimension.

Going further, you can also think about algorithms for determining the dimensionality reduction matrix of a manifold based on its tangent space."

Qiao Ze explained patiently, but as he spoke, although his speaking speed didn\'t change, and the speed of his hand writing formulas on the paper didn\'t change, his brow suddenly furrowed slightly.

Su Mucheng, whose attention was seven parts focused on the explanation and three parts on Qiao Ze, immediately noticed Qiao Ze\'s subtle movement and asked, puzzled, "What\'s wrong?"

Qiao Ze said, "I just realized that my thoughts had previously fallen into a trap, only thinking of constructing a framework with Group Theory. But in fact, thinking can be completely liberated, we can use a more concise mathematical approach to let machines accomplish the correspondence of invariant and variant mappings. I\'ve already got an idea."

"Right, then understand it following the process I\'ve written down. You only need to draw a few diagrams, compare the determinant algorithm with the matrix operation over time, establish a unit of area composition, and you\'ll understand the dimensionality reduction matrix."

Having said that, Qiao Ze then pushed the large piece of paper he had written on toward Su Mucheng, then looked up at Chen Yiwen, who was staring blankly at them, and said, "Chen Yiwen, you continue reading for now, if there are any questions we can talk about them later, I need to make some revisions to the framework I drafted this afternoon. Also, catch up quickly.

I think your work might be starting soon."

Afterward, Qiao Ze picked up his laptop and once again concentrated deeply on his research.

Chen Yiwen was utterly dumbfounded.

So this is the efficiency of self-study? Where on earth had his learning reached? Why was it that all the content seemed like something he hadn\'t even touched on yet?

Then he felt a bit wronged.

He was promised a problem-solving session tonight.

Never mind, sigh...

Darn gender advantage! May humanity perish!

As he thought this, Chen Yiwen still couldn\'t help but turn to Su Mucheng and ask quietly, "That... where have you self-studied linear algebra up to?"

"To the content of the second semester of sophomore year, I borrowed a textbook to follow along, why?"

"In just these few days, you\'ve already self-studied up to the content of the second semester of sophomore year?"

"Qiao Ze messed up the order of knowledge points, following his rhythm makes learning a lot more efficient."

Chen Yiwen nodded blankly.

There was such a method to learning?!

"Qiao Ze also said there\'s a method to solving math problems and a different one for research, not the same," Su Mucheng always spared no effort in praising Qiao Ze.

However, just as she finished speaking, she was corrected by Qiao Ze: "My actual words were that solving mathematical problems is very important, it can enable you to quickly grasp and understand what you\'ve learned, but if you want to do mathematical research, you must develop more flexible and open thinking on top of problem-solving."