Please, Translate the following English into Korean:
Alright, then let's jump into the next concept, which is called vector space. So I assume that you are interested in mathematics, you are interested in machine learning and then you can, nowadays, nowadays, you know, the vector representation is very important to construct the machine learning models to model the data or the real data, I mean, for instance, the text, image, the video or even the tabular data sets. Then at that time, usually we can say that from this vector representation, then we can say, what's the vector space and other kinds of things? And then that kind of concept is actually coming from the mathematics class, you know, the mathematics based on the linear algebra part. So please read these parts and understand the concepts for your research. Alright, anyway, so the vector space, so let's see the definition. So a real-valued vector space V, you know, so in this case, we are thinking about the real-valued vector space, then you can imagine that there is an imaginary-valued vector space or something like that, right? But it's kind of the abstract one, but as a practical point of view, so real value itself is enough. But anyway, so a real-valued vector space V, that one has these three kinds of elements or kinds of things, the V plus and then that, then that's the set V, so it's, you know, the V itself is set V with two operations. So here, the plus takes the two inputs, V and V, and then the output itself is V. And then that operation takes two things, but the first one is a real number and then the second one is V. Yeah, it's a real number, sorry about that. It's a real number. I mean, yeah, this one. And then the output of that operation is V itself. And then where, first, this one is an abelian group, yeah, abelian group, so the V and plus, those group is an abelian group. So that one is already defined in the previous slide, so you can easily understand the meaning of this one. And the second one is that it takes, I mean, there are several conditions. The first one is, as I said, is a V and plus is an abelian group. And the second one is that the distributivity, yeah, sorry about that. So it means that for all lambda, which is a real number, and X and Y, those are the vectors, right, in V, then lambda multiply X plus Y is the same as the lambda multiply X plus lambda multiply Y. Interesting. And then for all X, for all lambda and psi, it's the real numbers, and then X is the elements of the V, set V, and those operations, lambda plus psi multiplied X is equal to the lambda multiply X plus lambda, no, no, psi multiply X. Yeah, so that's the distributivity. And then with this V plus, and then that operation, that kind of the conditions are satisfied. And the second one is associativity. Associativity is actually, it's called as outer operation part. So there are, for all lambda and psi, which are real number, and X is the element of set V, then those things are defined. What's that? First, we just take the operation with psi and X and then lambda, then that's the same as the lambda multiply psi multiply or dot X, something like that. Yeah, so that's the associativity, right? That one is already, yeah, we mentioned in the previous slide, and I think you already know what's the meaning of this one. And then second, and the last one is the neutral element with respect to the outer operation. So here I said it's the outer operation part, right? Then we need to think about the neutral element. So neutral element means that it doesn't affect the output, right? So for all X in set V, one dot X is that, you know the X is that, right? So you know the dot itself is kind of the multiplication, then we can easily note that those are the neutral element of this operation. But anyway, yeah, sorry. So anyway, so what I wanted to say here is that I actually said that so these X is a vector, right? Already cause it's a vector space. But one of the interesting thing here is that, you know, starting from this to there, then actually, did I say like this one? No, yeah. I just say that. So this is just one of the set, and then those two things are at the operation. And then if that space is satisfied those conditions, then we can say that this V is a vector space, I mean, actually the real value of the vector space, and then one of the elements in V, I mean this one, yeah, now it's a vector. Huh, interesting. So in the previous time, I mentioned that the vector itself looks like, you know, the X is the maybe real numbers, and then it's a kind of groups, it's kind of the at least, right? So in the previous time, I'm saying about the definition of the vectors like that, but now in this slide, I didn't say like that, just, hey, there are the two operations and then without the operation and a set, then if that one is satisfied these kind of the conditions, then we can say that the elements of the set itself is called vector. Yeah. So that's the, sorry, that's the vector space, but anyway, yeah, that's sort of interesting, and then, you know, that one connects with each other. Well, that kind of phenomena itself is measured, observed in a probability area, and then I think you've already known that. So just imagine, I just think about the definition of the probability and then kind of the usage of the probability, then you might see some kind of difference between them, but at the end, they don't match each other. So with the same procedure, we usually think the vector itself is kind of the groups or list of the real numbers. However, we can also define the vector from the vector space, and then the vector space itself is defined as the set and then operations. Anyway, so with this kind of definition, then you can kind of easily understand the, what's the vector space.