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
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.
(전체 playlist: 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
1-2: Product of matrices (21:02)
https://www.youtube.com/watch?v=HDeVo4BSqFQ
1-3: Transpose of a matrix (16:56)
https://www.youtube.com/watch?v=wBa0E7r2w1g
Lesson 2: System of linear equations, Gauss elimination
2-1: Existence and uniqueness of solution (17:55)
https://www.youtube.com/watch?v=4Gm7rkHwx9I
2-2: Gauss elimination (44:26)
https://www.youtube.com/watch?v=ARpU9RIgoQE
Lesson 3: Rank of a matrix, Linear independence of vectors
3-1: Linear combination and linear independence (12:33)
https://www.youtube.com/watch?v=reoI81rndT4
3-2: Rank (31:01)
https://www.youtube.com/watch?v=j68sLmVZWEA
3-3: Using MATLAB (06:00)
https://www.youtube.com/watch?v=xIhjP-VeYXs
Lesson 4: Vector space
4-1: Vector space and its basis (26:23)
https://www.youtube.com/watch?v=D2k1fpQRHEA
4-2: Column space and null space (18:02)
https://www.youtube.com/watch?v=JGJzZ6e5ERA
4-3: Existence and uniqueness of solutions (19:02)
https://www.youtube.com/watch?v=hJJ-4NUB0-w
4-4: Vector space in general (19:09)*
https://www.youtube.com/watch?v=mLXLhcnwN9A
Lesson 5: Determinant of a matrix
5-1: Determinant (23:05)
https://www.youtube.com/watch?v=wE53sVqeHsM
5-2: Properties of determinant (40:04)
https://www.youtube.com/watch?v=F07WI0HXFRU
5-3: Cramer’s rule (15:19)
https://www.youtube.com/watch?v=KcwRKCcPdsU
Lesson 6: Inverse of a matrix
6-1: Inverse of a matrix (21:24)
https://www.youtube.com/watch?v=0ZcHCQ2NRP4
6-2: Gauss-Jordan elimination (19:22)
https://www.youtube.com/watch?v=QEmx-bsbeRc
6-3: Formula for the inverse (10:07)
https://www.youtube.com/watch?v=2WR3GvwAw_4
6-4: Properties of inverse and nonsingular matrices (18:05)
https://www.youtube.com/watch?v=e-lu0jq-A-k
Lesson 7: Eigenvalues and eigenvectors
7-1: Eigenvalues and eigenvectors (43:56)
https://www.youtube.com/watch?v=LAive9RvUQc
7-2: Examples (18:43)
https://www.youtube.com/watch?v=oArN3dhnHlk
7-3: Symmetric, skew-symmetric, and orthogonal matrices (24:59)
https://www.youtube.com/watch?v=Id0s1R2fq3I
Lesson 8: Similarity transformation, diagonalization, and quadratic form
8-1: Similarity transformation (09:37)
https://www.youtube.com/watch?v=aWU4BqdG3KY
8-2: Eigenbasis (17:52)
https://www.youtube.com/watch?v=SLicHtALNRc
8-3: Diagonalization (09:18)
https://www.youtube.com/watch?v=2mcPEEJxNLs
8-4: Jordan matrix and generalized eigenvector (21:30)*
https://www.youtube.com/watch?v=5aLpZDVDC1k
8-5: Quadratic form (15:00)
https://www.youtube.com/watch?v=xdUkJk6vlBw
Lesson 9: Introduction to differential equation
9-1: Function, limit, and differentiation (17:17)
https://www.youtube.com/watch?v=Hk-kiMXhugY
9-2: Differential equation, general and particular solutions (23:39)
https://www.youtube.com/watch?v=TVHCg37JmiI
9-3: Direction field, solving DE by computer (23:52)
https://www.youtube.com/watch?v=A82PShfJcj4
Lesson 10: Solving first order differential equations
10-1: Separable differential equations (20:30)
https://www.youtube.com/watch?v=PVFVr9Jza3o
10-2: Examples of separable DE (20:18)
https://www.youtube.com/watch?v=vLvXnYJP5bk
10-3: Exact differential equations (14:09)
https://www.youtube.com/watch?v=jopw5p1o3jA
10-4: Solving exact DE (31:28)
https://www.youtube.com/watch?v=cxaKWYcDHaU
Lesson 11: More on first order differential equations
11-1: Integrating factor (21:25)
https://www.youtube.com/watch?v=Gk1NcNkPo90
11-2: Linear differential equation (16:41)
https://www.youtube.com/watch?v=2jR7rmUzZ0s
11-3: Bernoulli equation (13:09)
https://www.youtube.com/watch?v=N_KffMg-e7U
11-4: Orthogonal trajectories of curves (08:51)
https://www.youtube.com/watch?v=JXg0j_yJwKA
11-5: Existence and uniqueness of solutions to initial value problem (25:28)
https://www.youtube.com/watch?v=Dt3IlRry2vw
Lesson 12: Solving the second order linear DE
12-1: Overview (10:57)
https://www.youtube.com/watch?v=zOWWM6n-bJc
12-2: Homogeneous linear DE (29:46)
https://www.youtube.com/watch?v=sK03Bj4Js7k
12-3: Homogeneous linear DE with constant coefficients (43:05)
https://www.youtube.com/watch?v=mXVrwPlvqyc
Lesson 13: The second order linear DE
13-1: Case study: free oscillation (24:40)
https://www.youtube.com/watch?v=Cyh6hDEtBic
13-2: Euler-Cauchy equation (24:09)
https://www.youtube.com/watch?v=CKTLavzjV3g
13-3: Existence and uniqueness of a solution to IVP (05:00)
https://www.youtube.com/watch?v=yfq9J89rZ2U
13-4: Wronskian and linear independence of solutions (28:23)
https://www.youtube.com/watch?v=-8Z50tu_2zs
Lesson 14: Second order nonhomogeneous linear DE
14-1: Nonhomogeneous linear DE and undetermined coefficient method (31:08)
https://www.youtube.com/watch?v=3jGuN4JKEdk
14-2: Examples (11:36)
https://www.youtube.com/watch?v=AqALTvhhZq8
14-3: Solution by variation of parameters (18:02)
https://www.youtube.com/watch?v=_0B_4qsoZMs
Lesson 15: Higher order linear DE
15-1: Higher order homogeneous linear DE (30:02)
https://www.youtube.com/watch?v=N0MnyyGi-xo
15-2: Higher order linear DE with constant coefficients and nonhomogeneous DE (23:34)
https://www.youtube.com/watch?v=fHfnctIcSOw
Lesson 16: Case studies
16-1: Mass-spring-damper system: forced oscillation (14:44)
https://www.youtube.com/watch?v=PCgmRsAy3EE
16-2: Mass-spring-damper system without damper (15:16)
https://www.youtube.com/watch?v=67-Cdd-HxSE
16-3: Mass-spring-damper system in general (20:38)
https://www.youtube.com/watch?v=AriXmB6eKro
16-4: RLC circuit (30:58)
https://www.youtube.com/watch?v=bJVnEUUvvMM
16-5: Elastic beam (07:10)
https://www.youtube.com/watch?v=fWUaCCW0s6w
Lesson 17: Systems of ODEs
17-1: Basic theory of systems of ODEs (17:13)
https://www.youtube.com/watch?v=3qgQwq3oyf4
17-2: Linear homogeneous case (12:20)
https://www.youtube.com/watch?v=trf71v6SwLw
17-3: Constant coefficient systems (26:14)
https://www.youtube.com/watch?v=DY-j6XoP7Yc
17-4: Constant coefficient systems: not diagonalizable case (15:57)*
https://www.youtube.com/watch?v=Zy0y6i_OgVE
17-5: Constant coefficient systems: more cases (11:22)*
https://www.youtube.com/watch?v=T7WGwiQdBtg
Lesson 18: Qualitative properties of systems of ODE
18-1: Phase plane and phase portrait (17:47)
https://www.youtube.com/watch?v=OuGR9ET5caY
18-2: Critical points (20:26)
https://www.youtube.com/watch?v=82YYrvh1lhE
18-3: Types of critical points (32:44)
https://www.youtube.com/watch?v=k_rmhiY4J34
18-4: Stability of critical points (15:45)
https://www.youtube.com/watch?v=XaPwm2aHIKI
Lesson 19: Linearization and nonhomogeneous linear systems of ODE
19-1: Linearization (20:57)
https://www.youtube.com/watch?v=oJF6s4YDbx4
19-2: Nonhomogeneous case (37:54)
https://www.youtube.com/watch?v=rnzGoSf9x7w
Lesson 20: Series solutions of ODE
20-1: Power series and radius of convergence (33:38)
https://www.youtube.com/watch?v=SxmdFjEXLDQ
20-2: Power series method (16:12)
https://www.youtube.com/watch?v=pguLLhPIJN0
20-3: Legendre equation (30:01)
https://www.youtube.com/watch?v=Lxl6ln8zbs8
Lesson 21: Frobenius method
21-1: Frobenius method and indicial equation (15:47)
https://www.youtube.com/watch?v=Hmw_7NtTyZc
21-2: General solution by Frobenius method (39:18)
https://www.youtube.com/watch?v=RGhAMxk4C6I
21-3: Example: Euler-Cauchy equation revisited (21:25)
https://www.youtube.com/watch?v=W-Saa-kwmrI
Lesson 22: Bessel DE and Bessel functions
22-1: Example of Frobenius method; a simple hypergeometric equation (25:37)
https://www.youtube.com/watch?v=nSgx54VIdH8
22-2: Another example of simple hypergeometric equation (10:46)
https://www.youtube.com/watch?v=HpsTet8dhXc
22-3: Bessel's equation and Bessel function of the first kind (19:39)
https://www.youtube.com/watch?v=9TxNkZkg5jE
22-4: Bessel function of the second kind and general solution (24:35)
https://www.youtube.com/watch?v=C93WDI5xm5Y
Lesson 23: Laplace transform Ⅰ
23-1: Introduction to Laplace transform (13:29)
https://www.youtube.com/watch?v=ZqEc24hIeT8
23-2: Linearity, shifting property (21:43)
https://www.youtube.com/watch?v=2mEvHuAtIsw
23-3: Existence and uniqueness of Laplace transform (18:34)
https://www.youtube.com/watch?v=BMatxMUvyMI
23-4: Computing inverse Laplace transform (09:08)
https://www.youtube.com/watch?v=_yyBbkqCNRQ
23-5: Partial fraction expansion & Heaviside formula (21:18)
https://www.youtube.com/watch?v=uFEWOmz6Q3k
Lesson 24: Laplace transform Ⅱ
24-1: Transform of derivative and integral (20:37)
https://www.youtube.com/watch?v=Xku1wxjrn0g
24-2: Solving linear ODE (15:42)
https://www.youtube.com/watch?v=jl8VZ-se42o
24-3: Unit step Function and t-shifting property (21:32)
https://www.youtube.com/watch?v=WMQbQ2d5ju8
24-4: Dirac's delta function (18:34)
https://www.youtube.com/watch?v=yMExWbSnKFw
Lesson 25: Laplace transform Ⅲ
25-1: Convolution (17:45)
https://www.youtube.com/watch?v=Zbb4ayJ4CCE
25-2: Properties of convolution (13:08)
https://www.youtube.com/watch?v=eI98RYWbaN0
25-3: Impulse response (21:23)
https://www.youtube.com/watch?v=Ro-5pLQvleY
25-4: Differentiation and integration of transforms (18:23)
https://www.youtube.com/watch?v=WoAc4BWrBxc
25-5: Solving system of ODEs (10:19)
https://www.youtube.com/watch?v=OyykwC3Q49Q