12. 쌍선형 형식은 여벡터-여벡터 짝(Bilinear Forms are Covector-Covector Pairs)

12. 쌍선형 형식은 여벡터-여벡터 짝(Bilinear Forms are Covector-Covector Pairs)



[0:00]In this video I want to introduce the idea of Covector-Covector pairs and show that Covector-Covector pairs are in fact the Bilinear Forms.



먼저 벡터와 여벡터 짝의 선형 조합인 선형사상에 대해 복습해보자. 벡터와 여벡터의 조합이 바로 텐서 곱이다(일반의 행렬의 곱이 아니다).

Recall in the last video I introduced new perspective on Linear Maps. "Linear Maps can be written as linear combination of vector-covector pairs" This process combining vectors and covectors together is called tensor product.



So, we have 'New Perspective' and this new perspective came along with more benefits. The first and most obvious benefits is we didn't have to remember the linear map transformation rules any more. We actually get them for free.

[0:53]So we write out linear map in the Old basis here[#1] and we wanna move to 'New' basis. All we have to do is we transform the basis vectors and basis covectors individually[#2]. So basis vectors are covariant, so to build old basis vectors out of new basis, we used the Backward transform, B and basis covectors are contravariant, so we build the old out of new using the forward transform, F. [#3] and we just pull this front[#4] and there we go [#5]. These are the components of linear map in the New basis. [#6]



So long as we know how to transform basis vectors and basis covectors, we get the transformation rules for linear maps almost for free.

기저 벡터와 기저 여벡터의 변환을 안다면 선형사상은 쉽게 구할 수 있다[기저벡터와 기저여벡터의 텐서곱의 선형조합이 바로 선형사상이다].



[1:33]Another BIG benefit is that when we have linear map acting on a vector, we can get the cracked matrix-vector component multiplication formula for free. We just have to replace the linear map on the vector[#1] with linear combinations expanded in some basis, and by using linearity rules[#2] and the Kronecker delta index cancellation rule[#3], we automatically get the cracked components for the output vector[#4] given by this multiplication formula.



[2:00]And finally the last benefit we get, I do not mentioned previous video. But, if we can consider tensor as arrays, the tensor product will automatically give us the cracked array shape. So, what do I mean by that?

[02:14] Well, before I show you that, one way of the column vector on the left multiply by row vector on the right, we obviously get a matrix as the result.[#1]




But there's another way getting this same result and that sometimes shown using this circled-times symbol in here. What this circled-times symbol is telling us to do is, telling us that take the array on the left and distribute it to each of the element inside the array on the right.[#2] So, this distribute that just means that we take this column vector here and give one copy to each element inside the second array here.[#3] And so we left with this which, if look at it, it's row of columns which basically like a matrix.[#4]

[2:56]So this way of thinking tells us that linear maps are row and columns. You might thinking in this is kind of stupid. Obviously linear maps are matrices. You don't need anyone telling you that and you are kind of right that this is a big surprise. But this idea of distributing arrays into each other will turn out to be very useful later on.

So, all of that is summary of how our perspective change on linear maps give us bunch of benefits.

지금까지 벡터와 여벡터의 조합(텐서 곱)으로 선형사상을 표현 했을 때 장점에 대해 살펴 봤다.
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측량 텐서를 포함해 쌍선형 형식은 여벡터와 여벡터의 선형 조합이라는 관점에서 살펴보기로 한다.

[3:22]Now next we're going through similar perspective change for Bilinear Forms. Of course it's including metric tensor. We're going to review Bilinear Forms as linear combinations of covector-covector pairs like this[#1]



So, first of all, why we're choosing covector-covector pairs to provide Bilinear Forms. Why not choose something else like vector-vector pairs. Well, Bilinear Forms take two vector inputs. Since covectors take one vector input each, a pair of covector would take two vector inputs. So, that's sort of hands that are on the right track covector-covector pairs. So, we just do the same process again. First of all we look at the transformation rules.



[4:00]If we assume we can write out any Bilinear Forms as a linear combination of covector-covector pairs to get the transformation rule for this components, or we just transform the basis covectors individually. So basis covectors are contravarient, so to build 'Old' from 'New' we use the forward transform, F. And putting this font, it gives us this transformation rule as you can see, cracked one. We can go to similar process for reverse transformation.
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[4:31]Next we can see, we can also get 'the cracked components multiplication formula' when Bilinear Form acts on two vector inputs.
[#1] So again all we do is replace the Bilinear Form and the vectors with their linear combination expansions in some basis. [#2]




[4:46]And what we'll do now is pass each of the vector inputs to their corresponding covectors. So the first vector is passed to first covector and second vector is passed to the second covectors[#3]. And by linearity of covector these come out in front[#4] and these become Kronecker delta and finally by the index cancellation rules[#5] we get this[#6]. And this is 'cracked component multiplication formula' that ends up giving us a single number as the result.
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[5:15]And finally the last benefit is the array shape. With Linear Maps we had circled-times operation with the column vector on the left and row vector on the right. But providing Bilinear Forms, since we dealing with covector-covector pairs, we're going to use two row vectors instead.



And again we're going to distribute array on the left to every elements of the array on the right. And we end up this which is basically row of rows. That might not seem right at first.
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[5:48]Recalling previous video I wrote out the array multiplication like this.[#1] where Bilinear Form or Metric Tensor was a Matrix. So, why did we get row of rows[#2]. Well, a row of rows actually makes a lot more sense. Remind this formula up here. Even though we have two vector inputs, we need to write one as a column and one as a row down at side to make them multiplication work correctly[#3].

That's kind of weird, because vectors should never be written as rows. That should always written as column, right?



[6:21]When we write Bilinear Form has row of rows, the matrix multiplication formula makes a lot more sense[#4]. We can write out both vectors as columns and you'll see that if we turn through this, we get the right answer. So, here is row and column. We just multiply them together like normally would, we do first times first, second times second. And now we have some of two rows here, all just add them together. And again we have a row and column here. So it is the same process, first times first, second times second. The final result is this which we could write as this summation. So, from an array multiplication stand point, a view in the Bilinear Forms as a row of rows actually makes a lot of sense.
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[7:08]Summarize what we've learned in this video. We learned Bilinear Form as linear combination of covector-covector pairs. And this immediately gives us the transformation rules. The components multiplication formula and the cracked array shape, all for free.



미궁속으로... 여긴 어딘가... 나는 누군가....
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[이전]11. 선형사상:벡터-여벡터 짝(Linear Maps are Vector-Covector Pair)
[다음]13. 텐서 곱 vs 크로네커 곱(Tensor Product vs. Kronecker Product)

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[구구단만 알아도 '텐서']
-1. 동기(Motivation)
0. 텐서의 정의(Tensor Definition)
1. 정역방향 변환(Forward and Backward Transformation)
2 벡터의 정의(Vector Definition)
3. 벡터 변환 규칙(Vector Transformation Rules)
4. 여벡터 란?(What's a Covector?)
5. 여벡터 성분(Covector components)
6. 여벡터 변환 규칙(Covector Transformation Rules)
7. 선형 사상(Linear Maps)
8. 선형사상 변환규칙(Linear Map Transformation Rules)
9. 측량 텐서(Metric Tensor)
10. 쌍선형 형식(Bilinear Form)
11. 선형사상은 벡터-여벡터의 짝(Linear-Maps are Vector-Covector Pair)
12. 쌍선형 형식은 여벡터-여벡터 짝(Bilinear Forms are Covector-Covector Pairs)
13. 텐서 곱 vs. 크로네커 곱(Tensor Product vs. Kronecker Product)
14. 텐서는 벡터-여벡터 조합의 일반형(Tensors are a general vector-covector combinations)
15. 텐서 곱 공간(Tensor Product Spaces)
16. 색인 올림과 내림(Raising/Lowering Indexes)

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11. 선형사상은 벡터-여벡터의 짝(Linear-Maps are Vector-Covector Pair)

11. 선형사상은 벡터-여벡터의 짝(Linear-Maps are Vector-Covector Pair)



지금까지 여러가지 벡터(vector), 여벡터(covector), 성형사상(linear maps), 쌍선형 형식(bilinear forms) 등 거의 모든 텐서를 다뤘다. 그 과정에서 좌표계의 변형에 대응하는 변환 규칙(transformation rules)에 집중했다. 지금부터 수준을 높여보자.



지금까지는 텐서의 종류를 나열하였는데 관점을 조금 바꿔 텐서를 다루는 법에 대해 논할 것이다. 이를테면 텐서를 가장 적절하게 정의한 '벡터와 여벡터가 결합된 텐서 곱'을 살펴보기로 한다. 지금까지 텐서를 다루던 방식(좌표계 변형에 따른 벡터 변환)과 아주 다른게 접근할 것이다. 텐서 곱을 배움으로써 텐서에 대해 좀더 폭넓은 이해를 가지길 바란다.



'텐서 곱'의 첫번째 강의로 벡터와 여벡터의 쌍(Vector-Covector Pairs)을 다룬다.



이번편부터 몇편에 걸쳐 텐서 곱을 다룰 텐데 이해를 돕기 위해 교과서나 여러 온-라인 문서에 나오는 표준의 텐서 곱 표기법에서 벗어날 것이므로 주의하기 바란다.



앞서 텐서는 '벡터와 여벡터의 조합'이라고 했다. 벡터와 여벡터는 텐서를 구성하는 기초단위다(building blocks). 이 두 기초 단위를 '텐서 곱'이라는 방법을 통해 생성한 것이 텐서다. 사실 선형사상이나 쌍선형 형식도 바로 벡터와 여벡터를 가지고 텐서 곱을 통해 생성된 것이다.



그럼 선형사상이나 쌍선형 형식이 어떻게 벡터와 여벡터로부터 만들어지는지 보자.

[이런 추상적인 수학을 영어로는 어떻게 설명하는지 들어보기로 하자. 실은 한글로 옮기기가 넘나 힘들다.]

[2:13]So we're already know that wen we multiply a row vector and column vector like this, a row vector first, we end up just scalar.



But we reverse the order, a column vector first, we'll get something different. Instead of scalar, it'll end up with matrix. And the entries of matrix will be these, here.



Ok, we have matrix and a matrix is basically linear map, right? So, there we go combination of vector and covector together to get a linear map.



So, this method combining row vector and column vector is sort of the first step to understanding tensor product. While we need to explore bunch of it before we can say what the tensor product is. So, we go back deeper here, i'm going to get you consider something.

[2:57] Let's have, we have this linear map here represented as matrix with the components; 4, 400, 8, 800. Now, can we go in the reverse direction? Can we figure out the components a,b,c,d that we break this matrix back of into column vector and row vector. So, you want to pause this video and try it by your self, but I'm gonna go ahead and give you answer. So, we have the column vector; 4, 8, and row vector; 1, 100. We can multiply them together to get this matrix, here.



Ok. Now consider this example where we have this same matrix, but the 800 has been changed to 1200. Can we do this same thing, break this matrix off into column vector and row vector? You can try how you'd like, but it turns out that it's impossible.



I'll prove that to you; the matrix elements have to equal to ac, bc ad and bd. And so following that reasoning; that means, ac equals 4, bc equals 8, ad equals 400 and bd equals 1200. The first thing we realize, these two [ac and bd], the first equation tell us that b equals to 2a, I'll just show here, it just follow this, we can get that b equals to 2a. We can apply similar reasoning, if we just substitute and follow the reasoning, we can show that b equals 3a. So, how can b equals 2a and 3a same time. That imply 2a equals 3a. That means that a has to be zero. This isn't obviously true. Otherwise entire first row of the matrix need to be zero. And it is NOT. So, this a equals zero is contradiction and it means it's impossible to solve for a, b, c and d, and such way that this column and this row multiply to get this matrix.



[5:00]What we've learned here is some matrix can be broken off into column vector and row vector and other matrix can't be. So we have two categories of matrices; Pure and Impure matrices. Pure matrix' components can be written as product of column vector and row vector component; whereas impure matrix component can't be written as this product.



Here are couple of examples of pure matrices and it turns out that pure matrices are actually really boring when they are used for Linear Maps. That's because all the output vectors exist along the same direction. The reason for that is ,as you notice what the pure matrix is, the columns of matrix are all scalar multiples each other. (4, 8) multiplied by 100 give us (400, 800), and (1/2, 1) multiplied by 2 gives us (1, 2).



We recall that since matrix' columns tell us where each basis vector copy goes, when it's put through Linear Maps, if all the matrix' column multiplies each other, that means that all the basis vectors give outputs that point in the same direction.  And that means that all possible vector inputs going through linear maps are same to the same direction. So, that's why pure matrices are kind of NOT very interesting set of transformation. They can do is really limited.

[06:35] But impure matrices are the more interesting ones. They can send basis vector to different actions. So we can get more interesting transformations.



So, we have a better problem, here. We can construct pure matrices using column vector-row vector products, but boring ones. How can we construct impure matrices, the interesting ones, using column vector-row vector product?



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[06:58] What I'm going to do is, I'm going to find 4-special vectors-covector pairs using old e-basis and old ε-dual_basis.



So, this matrix is 1 on top-left and zeros on what else. this can be written as the product which is really just product of basis vector e_1 and covector ε_1.



We can do the same things for the other basis vectors and covectors. We get (e_1,ε_1), (e_1,ε_2)(e_2,ε_1)(e_2,ε_2). We have 4-matrices, here.




[7:30] And you will notice that with these 4 matrices one take a linear combination. So one, we scale each matrix by different amount and add them all together, we can get any general 2x2 matrix that we like, just by packing the scaling members.



So, really this set of 4-products forms basis for matrices that a linear maps from the vector space V to itself.



So any general linear map, L can be written as this linear combination.



If we take the coefficients,.. and we can summarize this using the Einstein's notation, and see that any linear map, L can be written using this components, L of i,j.



Now you might be thinking; "All I see here is a vector and covector written in next each other, how can this be a Linear Map ?"

[8:24] Well, if we think of some linear map, L as a linear combination of this basis, linear maps and also have vector v which is linear combination of these basis vectors,


What we'll get when L act on input vector v ?

To do that we just substitute this for L and substitute this for input v. As you can see here, this ε^j, dual-basis vector, it's acting on input vector only, not totally. That's covector do there act on vectors.[#1]

So, we can use the linearity of ε to take out scaling coefficient v^k and put it on front, now we left ε^j acting on e_k.[#2]

Remember by definition, this is just Kronecker delta j,k.[#3]

And by the Kronecker delta index cancelation rule, we can cancel out the k index and replace it with j. [#4]

And finally, we get this.[#5] which is output vector written as linear combination of the e-basis vectors.

And these coefficients are just numbers, we can get from the standard matrix multiplication rule.[#6]

So, as you can see, this e, ε product pair, this really is linear map; take input vector, transform it and gave us output vector. So Vector-Covector pairs are really Linear-Maps. [#7]


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[9:50]Now, early here, I gave you this set of 4 linear maps and said that they form the basis all possible linear maps from V to V. But, just like the vectors there is nothing special about this basis. We can just as easily as choose another basis. Right? Even know those matrix looks very nice and simple, we can just easily take another basis matrices.



For example, those matrices here, they all form the basis for the set of 2x2 matrix. And that might pretty look very hard to believe, but in fact we choose the scaling numbers right, we can indeed get any 2x2 matrix that we want.



And likewise we don't have to choose those set of 4 linear maps['Old']. We can just as easily as choose these ['New'] set of 4-linear maps instead. So, these 'New' basis vector and 'New' dual basis covector pairs [for the basis for V-to-V map].



Then, now it be equally valid chore of basis.




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Let's sum up for this video. We've learned that we can combine vectors and covectors to get Linear Maps.

So doing with arrays, we put column vector first on the left[#1] and row vector second on the right[#2], and the result of that multiplication is matrix[#3].

And we can also do the things algebraically by just letting the vector, next to the covector like this[#4]. That is a Linear Maps that can take a vector as an input just like we showed before.

But the PROBLEM is that single vector and single covector combined together like this[#5] creates a PURE matrix or pure Linear Map, and those really boring. Because they send all the output vectors to the same direction.

[11:30]So to get the more interesting linear maps, we need to combine a bunch of pure linear maps take together in linear combination[#6]. That'll help us get more interesting IMPURE linear maps.[#7]



Introduce circled-times notation ⊗ for 'tensor product' symbol;



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[이전] 10. 쌍선형 형식(Bilinear Forms)
[다음] 12. 쌍선형 형식은 여벡터-여벡터 짝(Bilinear Forms are Covector-Covector Pairs)

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[구구단만 알아도 '텐서']
-1. 동기(Motivation)
0. 텐서의 정의(Tensor Definition)
1. 정역방향 변환(Forward and Backward Transformation)
2 벡터의 정의(Vector Definition)
3. 벡터 변환 규칙(Vector Transformation Rules)
4. 여벡터 란?(What's a Covector?)
5. 여벡터 성분(Covector components)
6. 여벡터 변환 규칙(Covector Transformation Rules)
7. 선형 사상(Linear Maps)
8. 선형사상 변환규칙(Linear Map Transformation Rules)
9. 측량 텐서(Metric Tensor)
10. 쌍선형 형식(Bilinear Form)
11. 선형사상은 벡터-여벡터의 짝(Linear-Maps are Vector-Covector Pair)
12. 쌍선형 형식은 여벡터-여벡터 짝(Bilinear Forms are Covector-Covector Pairs)
13. 텐서 곱 vs. 크로네커 곱(Tensor Product vs. Kronecker Product)
14. 텐서는 벡터-여벡터 조합의 일반형(Tensors are a general vector-covector combinations)
15. 텐서 곱 공간(Tensor Product Spaces)
16. 색인 올림과 내림(Raising/Lowering Indexes)

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10. 쌍선형 형식(Bilinear Form)

10. 쌍선형 형식(Bilinear Forms)



원래 '텐서곱(tensor product)' 강의로 이어질 계획이었으나 확실한 이해를 위해 몇편의 강의를 추가하기로 했다.먼저 텐서의 또다른 형태인 쌍 선형 형식(Bilinear Forms)이다. 앞선 강의에서 공부한 측량 텐서(metric tensor)가 쌍선형 텐서의 아주 특별한 경우다.



쌍선형 형식을 간략히 살펴보고 난 후 텐서곱을 다루기로 한다. 먼저 측량텐서를 복습해보자.

측량텐서는 기저벡터의 스칼라 곱이다. 스칼라 곱은 순서에 상관 없다. 이는 행렬에서 사선을 축으로 양쪽의 원소들이 일치 한다는(mirror images) 뜻이다.




그리고 '측량'이란 어떤 벡터의 길이와 두 벡터의 각도를 측정을 의미한다.




측량 텐서는 두개의 벡터를 입력받아 한 값(스칼라)를 출력하는 함수다. 따라서 두 벡터사이의 각도를 원한다면 두 벡터를 입력해 주고 길이를 원한다면 동일한 벡터를 입력으로 준다.




두 벡터를 입력으로 하는 측량 텐서의 계산식은 다음과 같다. 두 벡터를 다음과 같은 방식으로 연속적인 '행렬 곱셈'을 수행한다. [두 벡터중 하나는 행으로 다른 하나는 열로 놓고 행렬 곱셈 수행]



측량 텐서 함수의 a 배(scaling)는 함수 전체에 적용하거나 입력이 되는 각 벡터 혹은 벡터의 원소에 배값을 주어 계산한다.



주의 할 것은 배값을 두 입력인 벡터에 모두 적용해서는 않된다.



이번에는 덧셈 규칙이다. 함수 입력에 대한 덧셈 규칙,



하지만 입력의 분배 법칙 적용은 주의할 것.



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측량텐서는 쌍선형 형식의 아주 특별한 경우다. 쌍선형 형식은 측량텐서와 동일하게 두개의 입력을 받아 한 스칼라 값을 출력하는 함수다. 쌍선형 형식은 측량 텐서와 동일한 특성을 갖는다.


아울러 측량텐서와 마찬가지로 쌍성형 형식 또한 (0,2)-텐서다. 따라서 좌표계의 변형에 대하여 [벡터의 불변성을 유지하기 위한] 두개의 공변 규칙(행렬 곱)이 적용된다.



쌍선형 형식에 대해 알아보자. 먼저 '형식(Forms)'은 무슨 의미를 갖는지 알아보자. 복수의 벡터를 입력받아 한 실수를 출력하는 함수를 일컽는다. 앞서 배웠던 여벡터(covector)를 '선형 형식(Linear Forms)'이라 하는데, 입력에 대하여 함수가 선형성을 가지고 있으며 한개의 벡터를 입력받는 1-형식(1-Forms) 이라는 뜻이다.



쌍선형 형식(Bi-Linear Forms)는 두개의 벡터를 입력으로 받는 2-형식(2-Forms)이다. '쌍선형(Bilinear)'은 두 입력에 대하여 각각 독립적인 선형성을 갖는다.



두 입력 벡터 v와 w 중, v만의 선형성을 다룰 때 다른 입력 w는 상수 취급되어도 등식이 성립된다. 상대적으로 w만의 성형성을 다룰때 다른 입력 v는 상수 취급되어도 등식이 성립된다.

측량텐서와 쌍선형 형식모두 동일하게 특성들(두번의 공변 규칙적용과 선형성)을 가진다. 하지만 이 두 텐서는 확실하게 다른점을 가지고 있다. 앞서 말했듯이 측량텐서는 쌍선형 형식의 아주 특별한 경우라고 했었다. 다른 쌍선형 형식이 가지지 않은 측량텐서 만의 특성이라면 다음과 같이 두가지 점을 들 수 있다. 첫째, 입력의 두벡터 순서가 달라도 결과는 같다.  즉, g_ij와 g_ji가 같다. 행렬의 대각 원소를 기준으로 양쪽이 일치한다. 쌍선형 형식에서는 입력의 순서가 바뀌면 값이 달라진다. 두번째, 동일한 벡터를 입력으로 줄 경우 0보다 크거나 같아야 한다는 것이다. 벡터의 길이를 측정하기 위한 경우라 하면 길이는 0보다 작을 수 없다. 하지만 쌍선형 형식의 경우 음수가 나올 수 있다.



위의 예를 보자. 첫번째 측량텐서는 대각 원소 1과 1의 양쪽이 0으로 일치한다. 직교정규 좌표계에서 벡터의 길이 측정을 위한 텐서다[피타고라스 정리가 유효함]. 두번째 예는 대각으로 5와 5/16을 기준으로 양쪽 원소는 일치한다. 직각도 아니고 정규도 아닌 변형된 좌표계에서의 벡터 길이 측정 측량 텐서다. 이에 비해 비측량 쌍선형 형식의 예는 대각을 기준으로 좌우 원소가 다르다. 더구나 두번째의 경우, 만일 벡터 v의 원소가 1,1 이라면 출력이 -8로 음수가 된다[측량용이 아님, Non-metric]. 이 쌍선형 텐서는 측량 이외의 더 넓은 개념을 갖는다.


쌍선형 형식의 정의:

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[이전] 9. 측량 텐서(Metric Tensor)
[다음] 11. 선형사상은 벡터-여벡터 짝(Linear Maps are Vector-Covector Pairs)

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[구구단만 알아도 '텐서']
-1. 동기(Motivation)
0. 텐서의 정의(Tensor Definition)
1. 정역방향 변환(Forward and Backward Transformation)
2 벡터의 정의(Vector Definition)
3. 벡터 변환 규칙(Vector Transformation Rules)
4. 여벡터 란?(What's a Covector?)
5. 여벡터 성분(Covector components)
6. 여벡터 변환 규칙(Covector Transformation Rules)
7. 선형 사상(Linear Maps)
8. 선형사상 변환규칙(Linear Map Transformation Rules)
9. 측량 텐서(Metric Tensor)
10. 쌍선형 형식(Bilinear Form)
11. 선형사상은 벡터-여벡터의 짝(Linear-Maps are Vector-Covector Pair)
12. 쌍선형 형식은 여벡터-여벡터 짝(Bilinear Forms are Covector-Covector Pairs)
13. 텐서 곱 vs. 크로네커 곱(Tensor Product vs. Kronecker Product)
14. 텐서는 벡터-여벡터 조합의 일반형(Tensors are a general vector-covector combinations)
15. 텐서 곱 공간(Tensor Product Spaces)
16. 색인 올림과 내림(Raising/Lowering Indexes)

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