Name: Yuhai Oleksandr (유가이 올렉산드르)

Major: Bio-Mechatronic Engineering

Student Number: 2018

Ch. 1: Participation [참여평가] (10점)

(1) State more than 10 Math Definitions and concepts what you learned in Part 0, 1 and 2.

Part 0

(인공지능 개론)

Briefly reviewed the basic functions of Python, Sage and R, Basic concept of Power Method(거듭제곱법), Basic concept of MNIST 데이터셋을 활용한 손 글씨 숫자 인식(패턴인식), Briefly familiarized myself with the principle of work of AlphaGo…

Part 1

(인공지능 행렬과 데이터분석)

벡터, 내적, 벡터의 노름, 코시-슈바르치 부등식, 정사영(projection), 선형연립방정식(system of linear equations), 첨가행렬(augmented matrix), 기본행 연산(ERO), 행 사다리꼴(REF), Row Reduced Echelon Form, Gauss 소거법과 Gauss-Jordan 소거법, 행렬과 행렬식, vector sum, scalar multiplication, 행렬의 곱(product), 영행렬(zero matrix), trace, 역행렬, 가역행렬의 성질, 대각행렬, 스칼라행렬, 대칭행렬, 치환(permutation), 수반행렬(adjugate, adjunct), 일차독립, 일차종속, Subspace, 기저, 차원, nullity, column space and row space, 계수(rank), Rank-Nullity 정리, 최소재곱해(least square solution), Normal equation, Orthonormal basis, Gram-Schmidt 정규직교화 과정, QR 분해, 선형변환(Linear Transformation), 표준행렬, 핵(kernel)과 치역(range), 동형사상(isomorphism), 고윳값, 고유벡터, 대각화(Diagonalization), 닮음 행렬(similar matrix), 직교대각화(orthogonally diagonalizing), 고윳값분해(eigen-decomposition), SVD(특이값 분해), 일반화된 역행렬(Pseudo-inverse, Moore-Penrose Generalized Inverse), 이차형식(quadratic form), 주축정리(Principal axis theorem)…

Part 2

(다변수 미적분학과 최적화)

함수, 극한(limit), 도함수(derivative)와 미분(differentiation), 뉴턴 방법(Newton’s Method with gradient), 적분(Integral), 리만 합(Riemann sum), 외적(Cross Product), 벡터 함수, 편도함수(Partial Derivative)와 그래디언트(gradient), 연쇄법칙(Chain Rule), 방향도함수(directional derivative), Hessian, Taylor 정리, 함수의 극대(Local Maximum) and 극소(Local Minimum), Fermat’s theorem on critical points, 임계점(critical point), 안정점(saddle point), Gradient Descent Algorithm(경사-기울기 하강법), 중적분(double integral, multiple integral), Fubini의 정리, Polar Coordinates에서의 2중적분, Jacobian…

1. Python is an interpreted, high-level, general-purpose programming language. Python has become highly popular due to its abundance of libraries. Each Python library is application-oriented that was developed to address problems. Mathematical operations are most preferably carried out in Python due to its focus on utility and minimal programming jargon. There are several libraries that can be used to carry out mathematical operations with Python. Ex.: Math, Numpy, SciPy, Scikit-learn etc.

2. SageMath is a free open-source mathematics software system licensed under the GPL. It builds on top of many existing open-source packages: NumPy, SciPy, matplotlib, Sympy, Maxima, GAP, FLINT, R and many more. Access their combined power through a common, Python-based language or directly via interfaces or wrappers.

3. R is a programming language for statistical data processing and graphics, and is a free and open source computing environment under the GNU Project.

4. Power method is an eigenvalue algorithm: given a diagonalizable matrix A, the algorithm will produce a number λ, which is the greatest (in absolute value) eigenvalue of A, and a nonzero vector v, which is a corresponding eigenvector of λ, that is Av = λv.

5. MNIST(Modified National Institute of Standards and Technology) database is a voluminous database of handwritten numbers. The database is a standard proposed by the US National Institute of Standards and Technology for the purpose of calibrating and comparing image recognition methods using machine learning primarily based on neural networks.

6. AlphaGo is a "Go game" program developed by Google DeepMind in 2015. AlphaGo combines a technique that provided a major breakthrough in the power of "Go game" software in 2007-2012 (Monte-Carlo tree search) and recent advances in machine learning, namely deep learning with multilevel neural networks.

1. Vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its head.

2. Vector projection of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. It is a vector parallel to b, defined as: a₁ = a₁, where a₁ is a scalar, called the scalar projection of a onto b, and is the unit vector in the direction of b. Note that, scalar projection is defined as:

a₁ = ||a|| cos θ = a · = a ·

where · denotes a dot product, ||a|| is the length of a, and θ is the angle between a and b.

Note that, both the projection a₁ and rejection a₂ of a vector a are vectors, and their sum is equal to a, which implies that the rejection is given by: a₂ = a – a₁.

3. 기본행연산: 기본행연산법에는 3개의 방법이 있다.

· 두 행을 교환 (R_i ó R_j).

· 한 행에 0이 아닌 상수를 곱함 (kR_i ó R_i, where k ≠ 0).

· 한 행의 배수를 다른 행에 더함 (R_i + kR_j => R_i, where i ≠ j).

4. Gauss 소거법: 선형연립방정식의 첨가행렬을 REF로 변형하여 푸는 방법이다.

5. Gauss-Jordan 소거법: 선형연립방정식의 첨가행렬을 RREF로 변형하여 푸는 방법이다.

6. 행렬의 곱셈(세종대왕의 기역 법칙): 두 행렬 A = [a_ij]_mxp, B = [b_ij]_pxn에 대하여 A와 B의 곱(product) AB를 다음과 같이 정의한다.

AB = [c_ij]_mxn

여기서, c_ij = a_i1b_1j + a_i2b_2j + ··· + a_ipb_pj = (1 ≤ i ≤ m, 1 ≤ j ≤ n)

7. 가역 행렬(invertible matrix)은 그와 곱한 결과가 단위 행렬인 행렬을 갖는 행렬이다. 이를 그 행렬의 역행렬(inverse matrix)이라고 한다.

8. 대칭 행렬(symmetric matrix)은 전치 행렬(transposed matrix)이 스스로와 같은 행렬이다 (A = A^T).

9. 치환행렬(permutation matrix) 은 순서가 부여된 임의의 행렬을 의도된 다른 순서로 뒤섞는 연산 행렬이다. 일반적으로 치환행렬은 단위행렬로부터 얻을수있는 이진 행렬이다. 반대각행렬은 치환행렬의 특수한 경우이다.

10. 계수-퇴화차수 정리(rank-nullity theorem)는 행렬의 상과 핵의 차원의 관계에 대한 정리이다.

11. QR 분해(QR decomposition)는 실수 행렬을 직교 행렬과 상삼각 행렬의 곱으로 나타내는 행렬 분해이다. 그람-슈미트 과정이나 하우스홀더 행렬이나 기븐스 회전을 통해 얻을 수 있으며, 선형 최소 제곱법이나 QR 알고리즘에서 쓰인다.

12. 고유벡터(eigenvector)는 그 선형 변환이 일어난 후에도 방향이 변하지 않는, 영벡터가 아닌 벡터이다.

13. 고윳값(eigenvalue)는 고유벡터의 길이가 변하는 배수를 선형 변환의 그 고유 벡터에 대응하는 값이다.

14. 대각화 가능 행렬(diagonalizable matrix)은 적절한 가역 행렬로의 켤레를 취하여 대각 행렬로 만들 수 있는 정사각 행렬이다.

15. 고유값 분해(eigen decomposition)는 고유값 과 고유벡터로 부터 유도되는 고유값 행렬과 고유벡터 행렬에 의해 분해될수있는 행렬의 표현이다.

16. 특이값 분해(Singular Value Decomposition)는 행렬을 특정한 구조로 분해하는 방식으로, 신호 처리와 통계학 등의 분야 에서 자주 사용된다.

17. Quadratic form is a function on a vector space, specified by a homogeneous polynomial of the second degree in the coordinates of the vector.

18. Principal axis is a certain line in a Euclidean space associated with an ellipsoid or hyperboloid, generalizing the major and minor axes of an ellipse or hyperbola. The principal axis theorem states that the principal axes are perpendicular, and gives a constructive procedure for finding them.

1. Function(함수) is a binary relation over two sets that associates to every element of the first set exactly one element of the second set.

2. Riemann sum is a certain kind of approximation of an integral by a finite sum. One very common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations.

3. Partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant.

4. 기울기(gradient)란 벡터 미적분학에서 스칼라장의 최대의 증가율을 나타내는 벡터장을 뜻한다. 기울기를 나타내는 벡터장을 화살표로 표시할 때 화살표의 방향은 증가율이 최대가 되는 방향이며, 화살표의 크기는 증가율이 최대일 때의 증가율의 크기를 나타낸다.

5. Fermat's theorem is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point (the function derivative is zero at that point).

6. 경사 하강법(Gradient descent)은 1차 근삿값 발견용 최적화 알고리즘이다. 기본 개념은 함수의 기울기(경사)를 구하여 기울기가 낮은 쪽으로 계속 이동시켜서 극값에 이를 때까지 반복시키는 것이다.

7. Multiple integral is a definite integral of a function of more than one real variable, for instance, f(x, y) or f(x, y, z).

8. 야코비 행렬( Jacobian matrix)은 다변수 벡터 함수의 도함수 행렬이다.

(2) State more than 5 things that you know/can/find after you studied the first Part 1 and 2. (Please, note that, I will only indicate the most important terms(in my opinion) that I remember the most.)

1. After completing part 1, I can freely find the Inverse of the matrix. Below I will give in my opinion the fastest and simplest method for finding the inverse of a matrix. A method for finding the inverse of a matrix using Linear Row Reduction.

Step1: Adjoint the identity matrix to the original matrix.

Step2: Perform linear row reduction operations.

Step3: Continue until you form the identity matrix (on the left side of the newly augmented matrix).

Step 4: Write out the inverse matrix.

In Sage, let's say we have a square matrix A, to find its inverse we can use the command: A.inverse ().

2. Also, I can freely use Gram-Schmidt Process for ortho-normalizing a set of vectors in an inner product space. Imagine that we are given a basis {x₁, x₂, … ,x_p} for a nonzero subspace W or Rⁿ, so, we can define that,

v₁ = x₁

v₂ = x₂ -

v₃ = x₃ - -

⋮

v_p = x_p - - - ··· -

Then {v₁, … ,v_p} is an orthogonal basis for W. In addition, note that,

Span {v₁, … ,v_k} = Span{x₁, … ,x_k} for 1 ≤ k ≤ p

In Sage, we can easily applicate Gram-Schmidt process using the command (In the case of matrix A):

A.gram_schmidt().

3. I consider Singular Value Decomposition(SVD) to be one of the key topics of linear algebra, thanks to the professor and other students, I understood this topic more deeply. Let's go over the basic steps for SVD (Again, let's call our matrix A):

Step1: Compute transpose of our matrix, A^T, and also compute A^TA.

Step2: Determine the eigenvalues of A^TA and sort these in descending order, in the absolute sense. Then, square roots these to obtain the singular values of A.

Step3: Construct diagonal matrix Σ by placing singular values in descending order along its diagonal. Then, compute its inverse, Σ^-1.

Step4: Use the order eigenvalues from step 2 and compute the eigenvectors of A^TA. Place these eigenvectors along the columns of V and compute its transpose, V^T.

Step5: Compute U as U = AVΣ^-1. To compute the proof, compute the full SVD using A = UΣV^T.

In Sage, use the following code to calculate the SVD (an example is given for a 2x2 matrix, but you can use it for larger matrices, just slightly correcting the code in some places):

A = matrix([Given Matrix A])

B = A.transpose()*A

eig = B.eigenvalues()

sv = [sqrt(i) for i in eig] #특이값 구하기

print(B.eigenvectors_right()) #(A^T)(A)의 고유벡터 구하기, A의 right singular vectors

G = matrix([Matrix formed by Eigenvectors])

Vh = matrix([1/G.row(j).norm()*G.row(j) for j in range(0,2)]) #V의 전치행렬; Note: change range value based on matrix size

Vh = Vh.simplify() #V의 전치행렬 표현

print(Vh)

U = matrix([A*Vh.row(j)/sv[j] for j in range(0,2)]).transpose()

print(U) # U의 열들을 A의 left singular vectors

S = diagonal_matrix(sv)

print(S)

print(U*S*Vh)

4. I also fully understand and can use Newton's Method in optimization. Suppose you need to find the root of a continuous, differentiable function f(x), and you know the root you are looking for is near the point x = x₀. Then Newton’s method tell us that a better approximation for the root is

x₁ = x₀ -

Repeat this process as many times as necessary to get desired accuracy. In general, for any x-value x_n, the next value is given by

x_n+1 = x_n –

In Sage, you can use Newton's Method as follows:

#Use a loop to compute iterations of Newton-Raphson's Method

x = var('x')

f(x) = Given function

fp = diff(f,x)

NewtonIteration(x) = x - (f/fp)(x)

xn = Initial guess;

print('x0 =', xn);

for i in range(10): #Number of interations

xn = N(NewtonIteration(xn),digits=10);

print('x', i+1, '=', xn);

print("This is the exact answer for comparison: ")

print(solve(f(x)==0, x))

5. After studying part two, I fully understood the topic of multiple integrals. In particular, I understood well the meaning of Fubini's Theorem. Fubini's Theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value.

As a consequence, it allows the order of integration to be changed in certain iterated integrals. Fubini's theorem implies that two iterated integrals are equal to the corresponding double integral across its integrands.

In Sage, you can calculate multiple integrals using the following code (I will give an example code for calculating a double integral, but it can also be used for a triple integral or more):

f(x, y) = Given function

#in this example, I’ll integrate in the order dy dx

f(x, y).integrate(y, lower limit, upper limit).inegrate(x, lower limit, upper limit)

6. One of the most important topics in Part 2 is Gradient Descent Algorithm. Following are the main steps for using Gradient-Descent Algorithm.

Step1: Sets the initial guess and precision of the calculation , ε.

Step2: Then, calculate = – λ^[j] ∇F(), where λ^[j] = argmin_λF( - λ^[j] ∇F())

Step3: Finally, check the stop condition:

· If, | - |> ε; |F() – F()| > ε; or ||∇F()|| > ε (choose one of the conditions), then j = j + 1 and we need to go to step 2 again.

· Otherwise, = and stop.

Example Sage code from the textbook for Gradient-Descent Algorithm problem:

# initializing

A = diagonal_matrix(RR, [20, 10, 2, 1])

b = vector(RR, [1, 1, 1, 1]) # objective function

x0 = vector(RR, [0, 0, 0, 0]) # initial guess

g0 = -b # initial gradient

r = [] # 그래프를 그리기 위한 용도

# main iteration

for i in range(0, 200):

gn = g0.norm()

r.append((i, gn))

if gn < 10^(-8):

print("Stationary point! Algorithm terminated!")

break

w = A*g0

a = g0.inner_product(g0)/(g0.inner_product(w)) # step-size

x1 = x0-a*g0

g1 = A*x1-b

x0 = x1;

g0 = g1

show(line2d(r) + point(r, color = 'red')) # gradient 의 norm을 그래프로 그림

References:

[1] 이상구, 이재화., (2019), “인공지능을 위한 기초수학”, BigBook, Part 0, Part I and Part II

[2] David C. Lay, Steven R. Lay, Judi J. McDonald., (2016),“Linear Algebra and its applications”, Pearson, pp. 63~78

[3] James Stewart., (2016), “Calculus – Early Transcendentals”, Cengage Learning, pp. 10~270, 276~291; 345~348;

[4] Wikipedia., (2020), “Definition of some concepts from Parts 0, 1 and 2 above”, Wikimedia Foundation

· All the above pictures were used in accordance with the CC (Creative Commons) license.

(3) State your meaningful Comment/Answer/Discussions in Discussion/QnA. (Note: I will cite just a few of my responses, comments that I find really meaningful and helpful.)

1. Final by 천가영, 유가이 and SGLee [HW] 벡터에 관한 질문 by 천가영 <- 답변 by 유가이

1. 벡터는 크기와 방향을 갖는 유향선분 - 2차원, 3차원 공간의 벡터는 화살표로 표현 가능

-> Q1. 차원 벡터도 존재하나요?

답: 존재합니다.

만약 존재한다면 1차원 벡터는 화살표로 표현 불가능한가요?

답: 화살표로 표현하지 못하는 벡터들도 존재합니다)

아니면 1차원은 공간으로 간주되지 않아서 그런 것인가요?

답: 아닙니다. 1차원 벡터 공간도 존재합니다.

2. (R^n 안의) 벡터 X의 노름은 원점에서 점 P(x1, x2, ... , xn)에 이르는 거리로 정의됨을 의미한다.

->Q2 아래의 그림과 같이 시점이 원점이 아닌 벡터는 두 벡터의 합으로 밖에 표현할 수 없나요?

답: 아닙니다. 시점이 원점이 아닌 벡터는 원점으로 옮겨놓고 norm 노름을 구하시면 됩니다)

Remark by 유가이 올렉산드르: in answering this question, I tried to explain the concept of a vector in as much detail as possible. I also explained the concept of a vector in different spaces and gave several examples for a deeper understanding. Also thanks to the Professor's help (additional material and correction of minor inaccuracies), I hope that this answer cleared up all the student's doubts about the vector in space.

2. [Final OK by SGLee] Finalized by 한수현, 유가이 올렉산드르, 박지수] [HW-해공간을 구하는 법, Kernel(Null Space), Right Null Space, Left Null Space

Remark by 유가이 올렉산드르: Adding an additional answer and finalizing this post, I decided to tell more about Kernel(Null Space). I also explained the difference between right kernel and left kernel and used python for visualization. In the end I showed a slight difference when calculating null space in saga and python (using sympy library).

3. [Final OK by TA] [Final] Q by 김진웅 A by 유가이올렉산드르, 이상구 F by 김진웅 [HW 질문] 서로 같은 고유값에 대응하는 고유벡터들 차원 (JCF)

어떤 행렬 A 와 B 가 주어졌을 때 ...

서로 같은 (중근의) 고유값에 대응하는 (일차독립인) 고유벡터들이 이루는 부분공간 (eigenspace) 의 차원이 다를 때가 있는데 ...

어떻게 구분하여 알 수 있나요?

A 의 닮은 대각선행렬 :

B 의 닮은 대각선행렬 : 라는 의미는? 무엇인가요?

Here is my answer:

유가이올렉산드르(2018****33)7월 21일 ‎오전‎ ‎2‎:‎58

Let's try to analyze the given example.
The first matrix "D = ([[1, 0], [0, 1]])" has two eigenvalues (1 and 1) but they are obviously not distinct. Since A is the identity matrix, Av=v for any vector v, i.e. any vector is an eigenvector of A. We can thus find two linearly independent eigenvectors (say <-2,1> and <3,-2>) one for each eigenvalue.

But, the second matrix "D = ([[1, 0], [1, 1]])" also has non-distinct eigenvalues of 1 and 1. All eigenvalues are solutions of (A-I)v=0 and are thus of the form <t,0>. Hence, in this case there do not exist two linearly independent eigenvectors for the two eigenvalues 1 and 1 since <t,0> and <s,0> are not linearly independent for any values of s and t.

Remark 1: Every eigenvalue with multiplicity = n will be associated with n different (as in linearly independent) eigenvalues.

Multiplicity is how many "times" it shows up as an eigenvalue. It is like when you find only one solution to a second degree equation, which always has two roots. This solution has a multiplicity = 2.

Remark 2: If a matrix has more than one eigenvector the associated eigenvalues can be different for the different eigenvectors.

Geometrically, the action of a matrix on one of its eigenvectors causes the vector to stretch (or shrink) and/or reverse direction.

4. [Final OK by SGLee] Finalized by 나종진, 답변 by 유가이올렉산드르 , 직교대각화가능할 필요충분조건(교재 p.125) 질문 by 나종진

Remark by 유가이 올렉산드르: The student had minor problems with Sage code for Orthogonal Diagonalization, so I slightly corrected the code here.

My answer:

유가이올렉산드르(2018****33)7월 23일 오후 7:55

12번째 라인에서 C.row(i) => C.column(i)로 수정하시면 될 것 같습니다.

A=matrix([[0,3,3], [3,0,3], [3,3,0]])

print(A.eigenvectors_right())

print

x1=vector([1,0,-1])

x2=vector([0,1,-1])

x3=vector([1,1,1])

B=matrix([x1, x2])

[G, mu] = B.gram_schmidt()

print(G)

print y1=vector([1,0,-1])

y2=vector([-1/2, 1, -1/2])

C = column_matrix([y1,y2,x3])

P = column_matrix([C.column(i) / C.column(i).norm() for i in range(0, 3)])

print("P =")

print(P)

print

print(P.transpose()*A*P)

유가이올렉산드르(2018****33)7월 23일 오후 8:26

코드를 실행해봤는데 예제의 답과 같은 답이 나옵니다.

5. 답 by 유가이 올렉산드르, 이상구 교수님, 질문 by 한수현, Newton's Method에 대한 질문

두 설명(with and without gradient) 모두 Newton's method라고 하는 것을 보면, 각각이 결국 같은 방식을 이용하여 원하는 근사값을 구한다는 의미로 보입니다. 하지만 저는 저 두 내용이 완전히 관련 없는 독립된 내용으로 느껴집니다. 그래서 저 두 내용의 상관관계를 설명해주시면 좋겠습니다.

유가이올렉산드르(2018****33)8월 9일 오전 5:21

I think in both cases we are talking about the same iterative method (Newton's method). The difference is played by the dimension in which we work. As you know, in calculus, Newton's method is an iterative method for finding the roots of a differentiable function f, which are solutions to the equation f(x) = 0. In optimization, Newton's method is applied to the derivative f′ of a twice-differentiable function f to find the roots of the derivative (solutions to f′(x) = 0), also known as the stationary points of f. These solutions may be minima, maxima, or saddle points. In relatively Low dimensions we use classical Newton's method, but in Higher Dimensions we can also use Newton's method, all we need to do is replace the derivative with a gradient. also, please note that the geometric meaning in both cases remains the same. Also often Newton's method(with gradient) is modified to include a small step size 0 < gamma <= 1 instead of gamma = 1(pay attention to the classic formula, there the gamma is 1). Below I tried (taking into account the possibilities of the comments section) to give the basic formula that the Newtonian method represents (with and without a gradient). x_{k+1} = x_{k} - f'(x_{k}) / f''(x_{k}) and x_{k+1} = x_{k} - gamma[f''(x_{k})] / f'(x_{k}) The method is the same, just the area of application is slightly different. Newton's method with gradient is often done to ensure that the Wolfe conditions are satisfied at each step of the method. For step sizes other than 1, the method is often referred to as the relaxed or damped Newton's method. The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a paraboloid to the surface of f(x) at the trial value x_ {k}, having the same slopes and curvature as the surface at that point, and then proceeding to the maximum or minimum of that paraboloid (in higher dimensions, this may also be a saddle point). The central problem of optimization is minimization of functions, and Newton's Method copes with it relatively quickly. Hope this was helpful. Please correct me if I was wrong somewhere. [Reference: Read more on https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization]

이상구(LEE SANGGU)8월 9일 오전 6:57

Good. 같은 뉴턴 알고리즘 입니다. 차원만 인공지능 에서 사용하는 고차원으로 일반화 한것입니다.

(4) Below, I have consistently indicated all my QnA activity.

[1.1] [1st HW] 간단한 자기소개와 수강동기, 유가이올렉산드르

[1.2] [Final OK by SGLee] [수정본] 추가답변 및 Re-finalized by 유가이 올렉산드르, Finalized by 김범준 [1주차HW-질문] 일차독립

[1.3] [Final OK by TA] Final [HW (1주차 배운 내용 요약 1] 나종진 + 유가이 : 다항함수의 개형을 그리기 + [HW] 질문 Q4. 교재 p.110(PDF p.118)와 답 by 유가이올렉산드르, 나종진, 이상구교수님

[1.4] [Final OK by TA and SGLee] Final by 천가영, 유가이 and SGLee [HW] 벡터에 관한 질문 by 천가영 <- 답변 by 유가이

[1.5] Finalized by 유가이 올렉산드르 2-1강에 대해 질문 by 이상현, 답변 by 이상구 교수님

[1.6] [Final OK by SGLee] Finalized by 한수현, 유가이 올렉산드르, 박지수] [HW-해공간을 구하는 법, f Kernel(Null Space), Right Null Space, Left Null Space

[1.7] [Final OK by TA] [Final] Q by 김진웅 A by 유가이올렉산드르, 이상구 F by 김진웅 [HW 질문] 서로 같은 고유값에 대응하는 고유벡터들 차원 (JCF)

[1.8] [Final OK by TA] [HW] 열공간, 해공간, 영공간 사이의 관계, 질문 by 김진웅, 답변 by 장환승, 추가답변 및 Finalized by 유가이 올렉산드르

[1.9] [Final OK by SGLee] [HW] 1주차 강의 내용 요약 및 정리 + Sage과 Python 실습(비교), 내용 요약 및 정리 by 박진형, 코딩실습 및 Finalized by 유가이 올렉산드르

[2.1] [HW2] 선형변환 (Linear Transformation) 강의 요약 및 정리 + 연습문제 풀기

[2.2] [Final OK by SGLee] Finalized by 나종진, 답변 by 유가이올렉산드르 , 직교대각화가능할 필요충분조건(교재 p.125) 질문 by 나종진

[2.3] [HW2] 고윳값(eigenvalue), 고유벡터(eigenvector), 대각화(Diagonalization) 강의 요약 및 정리 + 연습문제 풀기

[2.4] [Final ok by SGLee] [HW2] 특이값 분해(Singular Value Decomposition, SVD) 강의 요약 및 정리 + 연습문제 풀기

[2.5] [Final OK by TA] 1주차 강의 내용 요약 및 정리 박진형, 코딩실습 및 Finalized by 유가이 올렉산드르

[3.1] [HW3] 함수(function), 극한(limit), 도함수(derivative)와 미분(differentiation) 강의 요약 및 정리 + 연습문제 풀기 + 코딩 실습

[3.2] [HW3] 미분의 응용 (Applications), 뉴턴-랩슨법 강의 요약 및 정리 + 근사 찾기 방법 비교 + 연습문제 풀기 + 코딩 실습

[3.3] [HW3] 적분 (Integral) 강의 요약 및 정리 + 연습문제 풀기 + 코딩 실습

[3.4] [Final OK by SGLee] [HW3] Finalized by 유가이 올렉산드르, 답변 by 이상구 교수님, Solved by 이상현 - 뉴턴 방법 증명

[4.1] [Final OK by SGLee] [HW 4] 연습문제 풀기 및 Re-Finalized by 유가이 올렉산드르, AI 을 위한 기초수학 16강 <미분적분학> [Gradient descent Algorithm] 개념 복습 및 정리 by 손원식, 추가 comment by 임성규, 이상구 교수님, 천가영

[4.2] Newton's Method에 대한 질문, 질문 by 한수현, 답 by 유가이 올렉산드르, 이상구 교수님

[4.3] Final OK by SGLee. [HW 4] 연습문제 풀기 및 Re-Finalized by 유가이 올렉산드르, Finalized by 손원식, AI 을 위한 기초수학 17강 <미분적분학> [중적분 (double integral, multiple integral)] 개념 복습 및 정리

[4.4] 경사하강법 실습(Newton's Method), 질문 by 한수현, 답 by 유가이 올렉산드르, 이상구 교수님, 김정한, 박진형

Ch. 2: Participation Part (참여부분, 정량) 자기 평가와 본인의 Project (Term paper) 제안서 등에 대해 아래를 채우시오. (20점)

Fill in the below for your self-assessment and your project/term paper. (20 points/100)

A. (Quantity 10pts) Briefly describe your contributions through Q&A for yourself and fellow students in our "DM" classes!

(1) Quantity: Check your participation numbers in Q&A <Refer your name in Q&A directly>: Each week (Saturday to Friday)

▪ Week 1: 9 Week 2: 5 Week 3: 4 Week 4: 4 (Total = entries) 22

Total number of sessions (Q: 7 , A: 15 )

■ Number of online attendances: ( 23 ) / 23 (1-4week)

Online WebEx OH attendance: ( 3 ) / 3 (1-4week)

Absences: 0

(2) 다음 밑줄 친 곳에 들어갈 내용을 고르시오.

나는 벡터, 내적, 벡터의 노름, 코시-슈바르치 부등식, 정사영(projection), 선형연립방정식(system of linear equations), 첨가행렬(augmented matrix), 기본행 연산(ERO), 행 사다리꼴(REF), Row Reduced Echelon Form, Gauss 소거법과 Gauss-Jordan 소거법, 행렬과 행렬식, vector sum, scalar multiplication, 행렬의 곱(product), 영행렬(zero matrix), 역행렬, 대각행렬, 스칼라행렬, 대칭행렬, 치환(permutation), 수반행렬(adjugate, adjunct), 일차독립, 일차종속, Subspace, 기저, 차원, nullity, column space and row space, 계수(rank), Rank-Nullity 정리, 최소재곱해(least square solution), Orthonormal basis, Gram-Schmidt 정규직교화 과정, QR 분해, 선형변환(Linear Transformation), 표준행렬, 핵(kernel)과 치역(range), 동형사상(isomorphism), 고윳값, 고유벡터, 대각화(Diagonalization), 닮음 행렬(similar matrix), 직교대각화(orthogonally diagonalizing), 고윳값분해(eigen-decomposition), SVD(특이값 분해), 일반화된 역행렬(Pseudo-inverse, Moore-Penrose Generalized Inverse), 이차형식(quadratic form), 주축정리(Principal axis theorem), 함수, 극한(limit), 도함수(derivative)와 미분(differentiation), 뉴턴 방법(Newton’s Method with gradient), 적분(Integral), 리만 합(Riemann sum), 외적(Cross Product), 벡터 함수, 편도함수(Partial Derivative)와 그래디언트(gradient), Jacobian, 연쇄법칙(Chain Rule), 방향도함수(directional derivative), Taylor 정리, 함수의 극대(Local Maximum) and 극소(Local Minimum), Fermat’s theorem on critical points, 임계점(critical point), 안정점(saddle point), Gradient Descent Algorithm(경사-기울기 하강법), 중적분(double integral, multiple integral), Fubini의 정리의 개념을 이해하고 설명할 수 있으며 간단한 것은 손으로, 복잡한 것은 sage 또는 python 을 이용하여 계산하여 그 의미를 설명할 수 있다.

(3) 개인/동료와 같이 “본” 강좌를 학습하면서 배우거나 느낀 점은?

I am very glad that I had the opportunity to once again get into this wonderful environment of the PBL class. I really wanted to deepen my mathematical knowledge in the field of Artificial Intelligence, and this summer semester, thanks to this subject (basic mathematics for artificial intelligence), I have this opportunity. As I said earlier, I really like the PBL technique, because thanks to it, each student can actively participate in the educational process.

(4) What you contributed through this course (Q & A and/or In-class):

I tried to actively participate every week in the QnA. Most of my QnA activity consists of answering questions, coding, finalizing and summarizing lecture materials. In almost every post I have tried to attach examples of solving problems by hand and using Sage or Python(or both) code, as I believe that a visual step-by-step process of solving problems in practice can greatly help me and other students in understanding the topic.

(5) (5 pt) What is your most important contribution or founding that you shared with others in QnA.(Quality)

To my mind, my most important contribution that I shared with others was:

[Final OK by SGLee] [HW] 1주차 강의 내용 요약 및 정리 + Sage과 Python 실습(비교), 내용 요약 및 정리 by 박진형, 코딩실습 및 Finalized by 유가이 올렉산드르

Here Mr. 박진형 summarized all the material for the first week, and I added Python and Sage code for deeper understanding the material. Also, my goal was to compare the capabilities of Python and Sage in solving mathematical problems. From this coding practice, we can conclude that it is much more convenient and faster to apply Sage in a mathematical environment.

(6) Number of Final OK by SG Lee Problems (and/or Completed Discussion/Question) in Q&A that your name is included: 13

(1) Write what you especially remember while you are doing A-1, 2, 3.

I am very pleased that most of the students in this class are very actively involved in the educational process. Everyone is trying to share their knowledge in QnA, ask questions on an incomprehensible topic, or help another student with a detailed answer. I also try to be a part of this process and to participate as much as possible in the QnA.

(2) What did you learn or feel while learning Math for AI (Action Learning/PBL) with your classmates?

I have greatly improved my mathematics knowledge and also deepened my knowledge of programming. Thanks to the active work of the Professor, TA and students, I significantly deepened my knowledge of SVD, Newton's Method and Gradient Descent Method and learned a lot about the way to apply them in practice.

Ch. 3: Personal Reflection Note (20점)

Subject		Basic Math for AI		Major		Bio-Mechatronic Engineering
Name		Oleksandr Yuhai		Year		2018 2nd Semester
Learning contents		Part0(Power Method, MNIST, AlphaGo). Part1(벡터, 선형연립방정식, 행렬과 행렬식, 일차독립과 기저(basis) 및 차원(Dimension), 선형변환 (Linear Transformations), 고윳값, 고유벡터, 대각화(Diagonalization), SVD (특이값 분해, singular value decomposition), 이차형식(quadratic form)). Part2(일변수함수와 미적분, 다변수함수와 미적분)
Self-Checking
Activity			Excellent		Good		Fair
1.	I have contributed to generate ideas and facts needed to resolve the issue.		●
2.	I proposed learning issues associated with learning.				●
3.	When I study alone, I used a variety of learning materials.		●
4.	I provide new information and knowledge in this class.				●
5.	I was actively involved in the discussions. And I provided a lot of questions in order to understand these discussions.				●
6.	I have made a contribution to the learning activities for our class.		●
※ Please record the following items by considering your learning process. 1. Do you understand the most of contents of this learning process? I understood at a sufficient level most of the topics from Part 0 to Part 2. I also learned many new methods of how to solve mathematical problems using programming languages (Sage, Python). 2. What kind of learning materials have you used to study? In the process of studying Mathematics for Artificial Intelligence, I most often used our professor's textbook and video lectures. But I also actively used various books such as Linear Algebra and its applications by David C. Lay, Calculus - Early Transcendentals by James Stewart, and others. I also used useful information from various math forums, thematic videos on YouTube, math blogs. I believe that all this was made possible through the use of the PBL learning system. Usually, I first get acquainted with the topic in our textbook book(Basic Math for AI by 이상구 교수님 with 이재화 TA님), then I watch the lecture of the professor, after that I look for more information on topics that are not entirely clear to me in other lectures of the professor (Linear Algebra, Calculus, etc.) on the site http: // matrix.skku.ac.kr/sglee/, after that for a deeper understanding of the material, I look for information in other books (Linear Algebra and its applications by David C. Lay, Calculus - Early Transcendentals by James Stewart, etc.). After that, I begin to search for more extensive information on the topic on the Internet (usually material related to practice). Finally, I start to practice solving tasks from textbooks manually and using Sage or Python (or both). After that, I usually post all the information I have collected in the QnA section and then start trying to answer questions from other students in QnA. 3. What did you learn through the learning activities of this course? Through learning activities I realized the importance of solving math problems not only by hand, but also using programming languages such as Sage and Python. I also understood the importance of active participation of students in the educational process. I believe that thanks to the active participation of students and Professor in the QnA, many even rather complex mathematical topics have become much clearer. Even though I have already taken courses on Linear Mathematics and Discrete Mathematics, I learned a lot of new things for myself from other students (for example, the scope of SVD). Also, since I did not take courses in Calculus, I learned a lot about such very interesting and important topics as partial differentiation, and the Gradient Descent Algorithm. 4. What have you learned from the other colleagues? From other colleagues, I learned a lot about Linear Algebra and Calculus, and thanks to other students, I learned many new ways to apply mathematical knowledge in practice. I also greatly improved my programming skills in the Sage language. I understood the importance of teamwork and mutual assistance to each other. Through this course I understood the importance of teamwork and mutual assistance to each other. 5. Self-Evaluation for Q/A Activities My score: 97 - Sometimes I posted my posts to the I-Campus a little late, as they took me a lot of time to write, but I tried to post at least 4 posts(summarizing the lecture materials, answering questions, finalizing, programming etc.) every week. 6. Evaluation for other students Almost all students in our class actively participated in the development of our Basic Math for AI class. I think that, thanks to the PBL system, many students were able to become a support for each other in understanding a variety of difficult mathematical topics. And I also think that thanks to the active participation of students, the professor, and TA in the learning process, many students were able to master most of the material from the first two parts at a fairly good level (even students from the Humanities and Social Sciences Campus who may not be familiar with Linear Algebra or Calculus before).

Self-Evaluation

Subject		Basic Math for AI	Major		Bio-Mechatronic Engineering
Name		Oleksandr Yuhai
Evaluation Items			Strongly disagree	Disagree		Mostly disagree	Mostly agree	Agree	Strongly agree
	1. I participated actively in both, online and offline classes.								●
	2. I participated actively on a Q&A activity.							●
	3. My question and replies made on Q&A are relevant.								●
	4. Information provided by my activity was useful for other students in the class.							●
	5. I enthusiastically took into the consideration other students’ opinions or point of view.								●
	6. I contributed to class by participating on Q&A discussions.							●
	7. I am enthusiastic about taking other class with the same students I am taking Basic Math for AI.								●
[Opinion] ► Satisfaction according to the Self-Evaluation I think that I was quite active in QnA, summarizing up the lecture material, practicing solving various problems manually and through the program code, finalizing the solutions of other students and, if possible, answering questions. Also, I think that, my work has helped other students to learn something new in solving problems in Linear Algebra and Calculus. ► Sorrow according to the Self-Evaluation Sometimes I posted my posts in the QnA section a little late (at the end of the week), but I tried to make each of my posted posts as informative and useful as possible.

Peer-Evaluation


Subject		Basic Math for AI
Colleague’s name		김범준
Name of evaluator		Oleksandr Yuhai
Evaluation Items			Strongly disagree	Disagree	Mostly disagree	Mostly agree	Agree	Strongly agree
	1. I participated actively in both, online and offline classes.							●
	2. I participated actively on a Q&A activity.							●
	3. My question and replies made on Q&A are relevant.						●
	4. Information provided by my activity was useful for other students in the class.						●
	5. I enthusiastically took into the consideration other students’ opinions or point of view.						●
	6. I contributed to class by participating on Q&A discussions.							●
	7. I am enthusiastic about taking other class with the same students I am taking Discrete Mathematics.						●
[Opinion] ► Satisfaction according to the Self-Evaluation A student from the Faculty of Consumer Science, pretty quickly adapted to the PBL system of this math related class and actively participated in QnA. I also believe that during his studies he improved his mathematical and programming (Sage) knowledge to a great extent. ► Sorrow according to the Self-Evaluation 없습니다.

Flipped/PBL Action Learning Class Reaction Evaluation(For Students)

Evaluation Details	Evaluation
Evaluation Details	Strongly agree	Agree	Neutral	Disagree	Strongly disagree
1. I participated actively in Flipped/PBL classes.		V
2. Professor helped students to demonstrate their abilities.	V
3. I obtained new skills and knowledge due Flipped classes.	V
4. Through that process of learning I developed my abilities of making hypothesis.		V
5. Through that process of learning I obtained or developed leadership capabilities.		V
6. I learned multiple methods of solving problems.	V
7. I am now able to use various materials needed to study efficiently enough.	V
8. I developed my professionalism.	V
9. The process of learning I experienced is similar to the real process of doing researches.	V
10. Evaluation of method of learning is relevant.	V