내용 : 선형대수학 + 상미분방정식 + 라플라스 변환
선형대수학(Linear algebra)이란 vector space와 vector space들 사이의 선형사상에 대한 수학이다. Vector space의 정의, 차원, 부분공간에 대해 배운다. 유한차원 vector space들 사이의 선형사상에 대한 선형대수는 행렬의 수학이다. 행렬의 행렬식(determinant)는 중요한 개념이다. 행렬식의 정의와 의미에 대해 배운다. Elementary operation과 가우스 조작에 의해 역행렬을 구하는 방법을 배운다. 행렬의 eigenvalue와 eigenvector에 대해 배운다. 대칭행렬, 직교행렬 등의 정의를 배우고 그 성질을 배운다. 복소수 행렬에 대해 교재에서는 다루고 있으나 이 부분을 수업에서 다룰 지는 결정되지 않았다.
여러가지 형태의 일계미분방정식의 풀이법을 배운다. 선형미분방정식의 해법에 대해 배운다. 선형미분방정식을 이해하기 위해선 선형대수에 대한 지식이 필요하다. 비선형 이계미분방정식중에서 Legendre 방정식과 Bessel 방정식에 대해 배운다. Sturm-Liouville방정식이론을 다룬다.
Laplace transform의 정의와 Laplace transform을 이용한 미분방정식의 해법을 배운다.
도움 : 서울대학교 CTL과 김준수 조교의 도움으로 만들었습니다.
2015년 2월에 서울대학교 교수학습센터의 도움으로 제작되었습니다.
심형보 교수, 서울대학교 공과대학 전기정보공학부, http://cdsl.kr/~hshim
Kreyszig, Advanced Engineering Mathematics (10판이 기준이나 다른 판도 내용이 유사하여 수강에 무리가 없음.), John Wiley & Sons, 2011
단, 동영상 청취에 따로 교재가 필요하지 않습니다. 대신 강의 노트를 화면 아래에서 PDF 문서로 다운 받을 수 있습니다. 혹은 https://s-space.snu.ac.kr/handle/10371/93973 를 참고하세요.
YouTube에서 가능합니다.
https://www.youtube.com/playlist?list=PL0Nf1KJu6Ui7yporXR_0CJwfovvllEgBk
Lesson 1: Introduction to matrix
1-1: Addition and scalar multiplication of matrix (17:33)
https://www.youtube.com/watch?v=K2JqVseszkQ&list=PL0Nf1KJu6Ui7yporXR_0CJwfovvllEgBk&index=1
1-2: Product of matrices (21:02)
1-3: Transpose of a matrix (16:56)
Lesson 2: System of linear equations, Gauss elimination
2-1: Existence and uniqueness of solution (17:55)
2-2: Gauss elimination (44:26)
Lesson 3: Rank of a matrix, Linear independence of vectors
3-1: Linear combination and linear independence (12:33)
3-2: Rank (31:01)
3-3: Using MATLAB (06:00)
Lesson 4: Vector space
4-1: Vector space and its basis (26:23)
4-2: Column space and null space (18:02)
4-3: Existence and uniqueness of solutions (19:02)
4-4: Vector space in general (19:09)*
Lesson 5: Determinant of a matrix
5-1: Determinant (23:05)
5-2: Properties of determinant (40:04)
5-3: Cramer’s rule (15:19)
Lesson 6: Inverse of a matrix
6-1: Inverse of a matrix (21:24)
6-2: Gauss-Jordan elimination (19:22)
6-3: Formula for the inverse (10:07)
6-4: Properties of inverse and nonsingular matrices (18:05)
Lesson 7: Eigenvalues and eigenvectors
7-1: Eigenvalues and eigenvectors (43:56)
7-2: Examples (18:43)
7-3: Symmetric, skew-symmetric, and orthogonal matrices (24:59)
Lesson 8: Similarity transformation, diagonalization, and quadratic form
8-1: Similarity transformation (09:37)
8-2: Eigenbasis (17:52)
8-3: Diagonalization (09:18)
8-4: Jordan matrix and generalized eigenvector (21:30)*
8-5: Quadratic form (15:00)
Lesson 9: Introduction to differential equation
9-1: Function, limit, and differentiation (17:17)
9-2: Differential equation, general and particular solutions (23:39)
9-3: Direction field, solving DE by computer (23:52)
Lesson 10: Solving first order differential equations
10-1: Separable differential equations (20:30)
10-2: Examples of separable DE (20:18)
10-3: Exact differential equations (14:09)
10-4: Solving exact DE (31:28)
Lesson 11: More on first order differential equations
11-1: Integrating factor (21:25)
11-2: Linear differential equation (16:41)
11-3: Bernoulli equation (13:09)
11-4: Orthogonal trajectories of curves (08:51)
11-5: Existence and uniqueness of solutions to initial value problem (25:28)
Lesson 12: Solving the second order linear DE
12-1: Overview (10:57)
12-2: Homogeneous linear DE (29:46)
12-3: Homogeneous linear DE with constant coefficients (43:05)
Lesson 13: The second order linear DE
13-1: Case study: free oscillation (24:40)
13-2: Euler-Cauchy equation (24:09)
13-3: Existence and uniqueness of a solution to IVP (05:00)
13-4: Wronskian and linear independence of solutions (28:23)
Lesson 14: Second order nonhomogeneous linear DE
14-1: Nonhomogeneous linear DE and undetermined coefficient method (31:08)
14-2: Examples (11:36)
14-3: Solution by variation of parameters (18:02)
Lesson 15: Higher order linear DE
15-1: Higher order homogeneous linear DE (30:02)
15-2: Higher order linear DE with constant coefficients and nonhomogeneous DE (23:34)
Lesson 16: Case studies
16-1: Mass-spring-damper system: forced oscillation (14:44)
16-2: Mass-spring-damper system without damper (15:16)
16-3: Mass-spring-damper system in general (20:38)
16-4: RLC circuit (30:58)
16-5: Elastic beam (07:10)
Lesson 17: Systems of ODEs
17-1: Basic theory of systems of ODEs (17:13)
17-2: Linear homogeneous case (12:20)
17-3: Constant coefficient systems (26:14)
17-4: Constant coefficient systems: not diagonalizable case (15:57)*
17-5: Constant coefficient systems: more cases (11:22)*
Lesson 18: Qualitative properties of systems of ODE
18-1: Phase plane and phase portrait (17:47)
18-2: Critical points (20:26)
18-3: Types of critical points (32:44)
18-4: Stability of critical points (15:45)
Lesson 19: Linearization and nonhomogeneous linear systems of ODE
19-1: Linearization (20:57)
19-2: Nonhomogeneous case (37:54)
Lesson 20: Series solutions of ODE
20-1: Power series and radius of convergence (33:38)
20-2: Power series method (16:12)
20-3: Legendre equation (30:01)
Lesson 21: Frobenius method
21-1: Frobenius method and indicial equation (15:47)
21-2: General solution by Frobenius method (39:18)
21-3: Example: Euler-Cauchy equation revisited (21:25)
Lesson 22: Bessel DE and Bessel functions
22-1: Example of Frobenius method; a simple hypergeometric equation (25:37)
22-2: Another example of simple hypergeometric equation (10:46)
22-3: Bessel's equation and Bessel function of the first kind (19:39)
22-4: Bessel function of the second kind and general solution (24:35)
Lesson 23: Laplace transform Ⅰ
23-1: Introduction to Laplace transform (13:29)
23-2: Linearity, shifting property (21:43)
23-3: Existence and uniqueness of Laplace transform (18:34)
23-4: Computing inverse Laplace transform (09:08)
23-5: Partial fraction expansion & Heaviside formula (21:18)
Lesson 24: Laplace transform Ⅱ
24-1: Transform of derivative and integral (20:37)
24-2: Solving linear ODE (15:42)
24-3: Unit step Function and t-shifting property (21:32)
24-4: Dirac's delta function (18:34)
Lesson 25: Laplace transform Ⅲ
25-1: Convolution (17:45)
25-2: Properties of convolution (13:08)
25-3: Impulse response (21:23)
25-4: Differentiation and integration of transforms (18:23)
25-5: Solving system of ODEs (10:19)