2. 線性代數 LINEAR ALGEBRA 2005 Spring

3. 教師及助教資料 o 教師：李程輝  Office : ED 828 ext. 31563 o 助教：葉易霖 林怡文  Lab: ED 823 ext. 54570

4. 參 考 資 料 o Textbook: S.J. Leon, Linear Algebra with Applications, 6 th Ed., Prentice Hall, 2002. o Reference: R. Larson and B.H. Edwards, Elementary Linear Algebra, 4 th Ed., Houghton Mifflin Company, 2000.

5. 成績算法 o Homework (10 %) o Middle Exam x2 (60 % each) o Final Exam (30 %) -------------------------- Total ： 100 %

6. Several Applications  How many solutions do have? It may have none, one or infinitely many solutions depending on rank(A) and whether or not.

7. o How to solve the following Lyapunov and Riccati equations: Matrix Theory

8. o Find the local extrema of definiteness of the Hessian matrix. o How to determine the definiteness of a real symmetric matrix? eigenvalues

9. o How to determine the volume of a parallelogram? Determinant

10. o How to find the solutions or characterize the dynamical behaviors of a linear ordinary differential equation? Eigenvalues, Eigenvectors vector space and linear independency

11. o How to predict the asymptotic( Steady-state) behavior of a discrete dynamical system ( p280.) Eigenvalues & Eigenvectors

12. o Given Find the best line to fit the data. i.e., find is minimum Least Square problem (Orthogonal projection)

13. o How to expand a periodic function as sum of different harmonics? ( Fourier series) Orthogonal projection

14. o How to approximate a matrix by as few as data? Digital Image Processing Singular Value Decomposition

15. o How to transform a dynamical system into one which is as simple as possible? Diagnolization, eigenvalues and eigenvectors

16. o How to transform a dynamical system into a specific form ( e.g., controllable canonical form) Change of basis

17. 課程簡介 o Introduction to Linear Algebra o Matrices and Systems of Equations  Systems of Linear Equations  Row Echelon Form  Matrix Algebra  Elementary matrices  Partitioned Matrices o Determinants  The Determinants of a Matrix  Properties of Determinants  Cramer’s Rule

18. o Vector Spaces  Definition and Examples  Subspace  Linear Independence  Basis and Dimension  Change of Basis  Row Space and Column Space o Linear Transformations  Definition and Examples  Matrix Representations of Linear Transformations  Similarity

19. o Orthogonality  The Scalar Product of Euclidean Space  Orthogonal Subspace  Least Square Problems  Inner Product Space  Orthonormal Set o Eigenvalues  Eigenvalues and Eigenvectors  Systems of Linear Differential Equations  Diagonalization  Hermitian Matrices  The Singular Value Decomposition

20.  Quadratic Forms  Positive Definite Matrices

21. Exercise for Chapter 1 o P.11: 9,10 o P.28: 8,9,10 o P.62: 12,13,21,*22,23,27 o P.76: 3(a,c),*6,12,18,23 o P.87: *18 o P.97: Chapter test 1-10

22. Exercise for Chapter 2 o P.105: 1,*11 o P.112: 5-8,*10-12 o P.119: 2(a,c),4,7,*8,11,12 o P.123: Chapter test 1-10

23. Exercise for Chapter 3 o P.131: 3-6,13,15 o P.142: 1,*3,5-9,13,14,16-20 o P.154: 5,*7-11,14-17 o P.161: 3,*5,7,9,11,13,15,16 o P.173: 1,4,*7,10,11 o P.180: 3,6-9,12,*13,16,17,19-21 o P.186: Chapter test 1-10

24. Exercise for Chapter 4 o P.195: 1,8,*9,12,16,18-20,23,24 o P.208: *3,5,11,13,18 o P.217: *5,7,8,10-15 o P.221: Chapter test 1-10

25. Exercise for Chapter 5 o P237: 6,7,10,*13,14. o P247: 2,9,11,*13,14,16. o P258: *5,7,9,10,12 o P267: 4,8,9,26,*27,28,29 o P286: 2,4,*12~14,16,19,22,23,25,33 o P297: 3~5,12,*4 o P310: Chapter test 1~10

26. Exercise for Chapter 6 o P323 : 2~16, 18, *19, 22~*25, 27 o P351 : 1 (a)*(e), 4, 6, 7, 9~12, 16~18, 23(b), 24(a), 25~28 o P363 : 8, 10~*13, *19, 21 o P380 : *5, 6 o P395 : 3(a)(b), 7(a)(b), 8~14, *12 o P403 : *3, 8~13 o P421 : Chapter test 1~10