MATH 410 SUMMER 2004 (Prof. Bayly): SYLLABUS (tentative):

The timing may vary from this proposed timetable. Make sure to stay up to date with the course as it actually evolves. The section headings refer to sections in Olver and Shakiban; topics marked with an asterisk* are not covered in O&S (yet!).

12 July: 1.1, 1.2, 1.3, 1.8. Review of vectors, matrices, linear systems. Geometric interpretations. Gaussian elimination, echelon forms. Existence and general form of solutions of linear systems.

13 July: 1.2, 1.3, 1.4, 1.5, 1.6. Vector and matrix operations, transposes. Matrix optics application*. LU factorization, matrix inversion. Row and column exchange issues. Rank and nullity of matrix and transpose. Null vectors as “cycles”*.

14 July: 2.1, 2.3, 2.4, 2.5. Vector spaces, subspaces, linear independence, basis and dimensions. Four fundamental subspaces. General theory of existence and/or uniqueness of solutions.

15 July: 2.6, 6.1, 6.2, 6.3. Applications and special matrices. Networks, circuits, bridges and trusses. Review for exam 1.

16 July: Applications continued, currency arbitrage*. In-class review for exam 1.

19 July: 3.3. Norms, vector and matrix max norm. Invertibility of near-identity and diagonally-dominant matrices, Jacobi iteration. EXAM 1 on material in chapters 1, 2, 6.

20 July: 3.1, 3.2, 3.4, 3.5. Inner products, length (2-norm). Quadratic forms and symmetric matrices, Cholesky factorization, positive-definite and other classifications. Maxima, minima, saddle points. Gram matrices.

21 July: 4.1, 4.2. Minimization of multivariable quadratic functions. Minimal length solutions of underdetermined linear systems*, quadratic and linear formulations.

22 July: 4.3, 4.4, 4.5. Least-squares approximate solution of overdetermined systems, normal equations. Applications to data-fitting, function approximation, and ranking of teams.

23 July: 3.2, 5.6. Angle between 2 vectors, orthogonal vectors. Orthogonal complement of a set of vectors. Orthogonality properties of fundamental subspaces of a matrix. In-class review for Exam 2.

26 July: 5.5. Projections parallel and perpendicular to vectors. Least squares approximation as projection onto column subspace. Projection matrices. EXAM 2 on material in chapters 3, 4.

27 July: 5.1, 5.2, 5.3. Orthogonal and orthonormal matrices, Gram-Schmidt orthonormalization, QR factorization and algorithm. Areas, volumes, hypervolumes.

28 July: 1.9. Determinants and applications. Signed areas, volumes, and hypervolumes. NOTE: we will completely skip chapter 7! This chapter presents a more abstract view of the preceding 6 chapters, but as O&S write on page ix of the preface, this material can be omitted if time is an issue.

29 July: 8.1, 8.2, 8.3, 8.4, 8.5. Eigenvalues and eigenvectors, and their properties. Characteristic polynomial. Diagonal form of a matrix (mention Jordan form). Special properties of symmetric matrices. Rayleigh quotients.

30 July: 8.6. Singular Value Decomposition. In-class review for exam 3.

NOTE: Chapters 9 and 10 develop various aspects and applications of eigenvalue-eigenvector theory, and the order in which you do them is not very important. I will do it in a different order than O&S because I think it fits slightly better with UA’s Math410 but you should not read much significance into either their or my choice of topic order.

2 August: 10.3 Eigenvalues and norms, spectral radius, Gershgorin Disk Theorem. EXAM 3 on material in chapters 5, 8.

3 August: 10.1, 10.2, 9.6. Powers and polynomials of matrix, Cayley-Hamilton theorem, exponential of matrix.

4 August: 10.2, 10.4, 10.5. Linear discrete-time systems. Fibonacci-like recursions, Markov systems, network models. Long-time behavior. Jacobi iteration (again), Successive Over-Relaxation.

5 August: 9.1, 9.2, 9.3, 9.4. Linear continuous-time systems. Solution to 1^st-order homogeneous systems in terms of matrix exponentials, general method for matrix exponential. Continuous-time Markov systems.

6 August: 9.5. Mass-spring systems and higher-order DEs. Vibrational modes, Rayleigh quotients (again).

9 August: Course review.

10 August: Course review.

11 August: FINAL EXAM half on chapters 9, 10 material, half comprehensive.