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Okay, so thank you for coming this video. This video is so we are talking about the analytic geometry part. So the first concept in this video is norm. So a norm on a vector space V itself is a function that takes the vector which is in a vector space right and then there are maps to the real number R and then that denotes as this kind of the shape. And then we can know that so that one actually assigns each vector x its length. Yeah, then it denotes like that. Such that for all lambda which is a real number and then x comma y, those are the vectors in a vector space V, then that three properties are far and the whole. So the first one is the absolute homogeneous one. You know, you know, they are here the lambda x and the lambda itself is, you know, the real number and then it's a kind of the coefficient of the vector x. And then that lambda can be go out of this norm. But you know, the lambda itself should be the absolute norm. So if you know the absolute homogeneous one, then you can know that the lambda is the real number. That means that, you know, the lambda should be greater than equal to zero. Yeah, even and then the. Yeah, and then. Oh, yeah, so I need to write that. Okay, so I guess you already know what's the meaning of the absolute variable. Let's define it. So lambda. Yeah, if. Sorry. Lambda. Is greater than equal to zero. And negative lambda. If lambda is negative value. All right. That's the definition of the absolute value of lambda. All right. So here the absolute homogeneous means that lambda x norm of the lambda x is the same as the norm of x multiplied the absolute value of lambda. All right. And then the second one is the triangle inequality. So the triangle inequality is the norm of x multiplied by the absolute value of lambda. So this is the triangle inequality. So I think you've already heard about this property in a Euclidean space. All right. So and then at that time and then I also said that the norm itself could be thinking about the length of the vector. So the x plus y vector's norm is less than equal to the. Oh, that's. The norm's. Summation. So they don't actually shows like that. So here there are kind of two vectors. And then the A is right there. And then the B is like there. And then the C. That one is actually what's that. Yeah. So here that's this ABC is our length. But anyway, so this. This is the length of the vector. So that's the length of the vector. And then the C. That one is actually what's that. One. For example, this is a vector x. And then this is a vector y. And then we know that from here to there. Then that's the x plus y. And then here then the norm of x. Sorry. Yeah. The norm of x is A. And then the norm of y is B. And then. The norm of x plus y is. C. Then we already know that C that one is less than equal to A plus B. And then when the equality is meets. So that's quite. That's the one of the questions. Yeah. In a case. All right. And then. And then. The next property is positive definite. So that means that the norm of x is greater than equal to zero. And the norm of x itself is zero. And that means that the x itself is a zero vector. Yeah. So here the n means that the norm should be greater than equal to zero. And there is no negative norm value. Yeah. Yeah. And then. You know. If the norm of x itself is zero. Then that means that the x is a zero vector. All right. So that's the property of norm. And then here there are several norms. In this area. And then I will introduce the two specific norms. The first one is Manhattan norm. And then. The Manhattan norm denotes like that. The norm of x. And then write one. That's because. When we wanted to calculate this norm vector. Then they don't define like that. So here x is a vector. And then the x i is element in a vector. And then here we know that x is an n dimensional vector. So we just add the. Add the. Add the. Add the. And then the x is the absolute value. Of the elements of x. And then the summation of those whole absolute values. Then that's the Manhattan norm. And then when we think about Manhattan norm is equal to one. Then they don't reach to this kind of square. So, you know. This vector. This vector. And then this vector like that. And then. And then. And then the second one is the Euclidean norm. So the Euclidean norm is that there are denoted as two, and then, you know, that's the definition itself is like that. The square root of the summation i to the n, xi squared, right? And then we know that that one actually denotes like that, the square of x transpose x. Because here, the x itself looks like a step, the column vector looks like a column vector, x1, x2, ta-ta-ta, xn, right? So that's the n-dimensional space, n-dimensional vector 1. So that's why, so when we compute this x transpose x, then x1, x2, ta-ta-ta, xn, and then x1, x2, ta-ta-ta, xn. And then, you know, the top product is, come to like that, x1 multiply x1 plus x2 multiply x2. That's the same as this expression. And then when we see that, then here, we can also draw some kind of circle from the origin, right? So these two things are quite different. One, and then, please remember these two specimens, we've already known and they are asking us about the model recurs. What is that? It's called a wanthan norm. And actually, I think you've already known what this part, so that you kill a norm, right? the right and then the go to south something like that so that's why that kind of a structure is called marathon and then here this kind of the expression is called marathon norm okay and then the next concept which we wanted to see which we wanted to see is dot product so the definition of dot product actually I won't say about it but anyway so the dot product is those things so here there is a kind of n dimensional vector x and y so x is x1 x2 ta ta ta xn and then y is also the shape looks like this you know that's the n dimensional vectors and then when we wanted to compute the x transpose y then you know x1 x2 ta ta ta xn and then y1 y2 ta ta ta yn and then this direction so we know that this one actually the same as this expression the xi multiply yi and summation of those those whole whole things so that's also that's that's kind of the you know the linear combination but here the X and Y are the vector one all right so that kind of the product is called dot product and then actually this is a very popular one in a real space kind of real space and a real situation we are in an environment not that and then let's move on to the kind of general concept and then here let's see a omega that one is a bilinear mapping one means that the omega and it takes the two components two objects as an input right and here the first one is lambda x plus psi y psi y and then the second input is z and then we know that x y z those are vectors then when we take the omega for these two inputs then the output is looks like this so lambda multiply omega x comma z plus psi omega y comma z so that means that so we z that one is combined into x and y separately and then we can extract the lambda and psi outside of the um operator, omega1. With the same procedure, and then here, the first input is x itself, and then second input is lambda y plus psi z, then here lambda is go out, and then psi is also go out since lambda and psi, those are real numbers, and then the omega takes two objects, two inputs, x and y, and here, and x and z. Alright? So that's the that's just kind of the bilinear mapping one. And then with this bilinear mapping, then we can say that this omega, I mean bilinear mapping omega is symmetric if omega is the same as omega . So we can change the order of the inputs. That means that the omega doesn't care about the order of inputs. Alright? And then second, kind of the important property is that we can recall that omega is positive definite if, if for all x in v except zero vector, omega is greater than zero. And then omega , so you know, those same thing, but here the general vector, then the output is zero number. If the omega is satisfied with this condition, then we say that it is positive definite. So please remember those two properties, so the symmetric and the positive definite. So these are very popular and then important concept to understand the several operators or the matrices in a later. So please remember and please memorize these two properties, or the kind of condition. And then here, let's define an inner product. So the inner product itself is a positive definite and symmetric bilinear mapping omega. So here, when we say about the omega, I mean which is the bilinear mapping, then that one satisfies these properties. That's why it's called bilinear one. And then this omega satisfies the two conditions, symmetric and positive definite, then we call this omega as inner product. So the omega , then we just denote the inner product like that. The and then it looks like this. So from now, when I write this kind of thing, that means that it's an inner product. Alright? And then actually, so that product is a kind of the inner product. So inner product is sorry for my wrong not good writing. So that's the important concept you need to memorize. Usually, most of the times, we just say that the inner product is the same as the dot product, and then but strictly speaking, it's not. So we can say that dot product is inner product, but there are another product, another mapping function that satisfy this condition. The positive definite and symmetric bilinear mapping one. So one of the example here is I've, that one is denoted actually in textbook. So x1, y1, negative x1, y2 plus x2, y1 plus 2x2, y2. If the x and y, those are the real, the two dimensional vector one. Then, you know, there is also kind of the operator, right? And then, it actually positive definite symmetric bilinear mapping. But we know that this is not a dot product. Right? Because the dot product is that the summation, for this case, the R square case, then dot product is that product on the two dimensional space is the x1, y1 plus x2, y2, right? So that's not a same thing. Here, this one is also the inner product. And actually the dot product itself is the inner product. But I wanted to say that there is another product that satisfy, sorry, there are several, there is other, sorry, there is other products that are inner product, but it's not a dot product. They are not dot product. Alright. So, as I said, so that's the definition of inner product, and then let's see some kind of a property of inner product. Alright. So, here this is inner product, and then when we use the basis, alright, bi. So here the x and y, those are any vectors in vector space v, and then, then actually this is the n dimensional R space. Then, you know, using the one of the basis, v1 to vn, then we can denote the x and y like that. And then this one actually change to this kind of format. Since, you know, inner product is what's that? Bilinear mapping one, so satisfying these properties, right? So using that one, then we can denote this inner product like that, and then we actually change the format like that. Where A is, matrix A is looks like this. So Aij is the inner product of vi comma vj, and then x hat and y hat, those are coordinates of x and y with respect to the basis bi. Yeah, so that's quite interesting point. And then, actually, so one of the interesting things is that asymmetric matrix A itself is R and n, that satisfy the following condition. For all x, which is a vector space in a vector v, but except the zero vector, then x transpose A x is greater than zero. So here, let's see this one, so it's kind of inner product one, right? And then that one change to this kind of format, and then if the symmetric matrix here, the symmetric matrix A, and then we take the x transpose x, and they are always greater than zero, where x is any vector except zero vector, then we call this kind of vector as symmetric positive definite matrix. So please remember this definition, and that one is actually usually useful for machine learning, for constructing machine learning, and for constructing and understanding machine learning, and then machine learning algorithm itself. So please remember this one right now, and then I will also describe the kind of application of this matrix in a later. So, and then I said that, so that kind of thing is actually related to the inner product one, and then usually we can say that that product is one of the inner products, so they don't actually relate to these kind of things. And then that product itself is also similar, which is related to the norm, and something like that. Okay, yeah, that's it, so please remember this concept, and we will see it later. Alright, and then the next concept which you wanted to know is Cauchy-Schwarz inequality, and I assume that you've already know about this thing, so in a vector space case, so then the absolute value of the inner product those are the less than equal to the norm of each one, the multiplication of the norm of each vectors. Because of time, so I don't have any time to describe about this one, and then the proof of this Cauchy-Schwarz inequality. But I will also, I will provide additional materials that proves this inequality. So please watch that one, and then understand this inequality, because that one is very important and very popular inequality in the future, and the future means the generator. Alright, and then the last concept in this video, I wanted to introduce in this video is vector distance. So here, we wanted to compute the distance between two vectors for instance, x, y, x and y, then Theron denotes as like that, that d is a kind of function and then takes two inputs, x and y, the definition of d, x, y, which is the vector distance is the norm of x minus y. And then we already know that the norm itself looks like this, the square root of the inner product of those two things, x, y. Here, x, y, this is also a one of the vector, right? So we can compute the inner product of this jet, and then we can take the square root of that. And then one of the interesting is that the d and there are several interesting points, the first one is that d is a positive definite, and the second one is that d is symmetric. So the positive definite means that, you know, the d , sorry, the d , that's the greater than or equal to zero, right? And the second one is that d , the symmetric means that d , right? And the third one is that it also satisfies the triangle inequality. For instance, there are three, what's that? Three vectors, and then d , that's the less than or equal to the d plus d , not x, sorry, y, z. Yeah. So that's the triangle inequality, and then those properties are satisfied when we say about the vector distance. Why, then if you wanted to know the reason, then please follow the definition, and then the vector, the definition of these properties, and then also definition of the norm. Then you can easily find the reason for that in the next video. Alright, so that's the all for this, that's the all in this video, and then we will see another, another kind of concepts in the next videos. Alright, so thank you for watching this video, and then see ya.