MATH 410 SUMMER 2002 (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.
8 July: 1.1, 1.2, 1.4, 1.6.
Review of vectors, matrices, linear systems.
Geometric and algebraic interpretations. Vector and matrix
operations, transposes. Lengths of vectors. (HW: 1.2(4, 8, 9, 11), 1.3(2, 12, 13),
1.4(2, 3, 22)
9 July: 1.3, 1.5, 1.6, 2.2. Gaussian elimination, LU factorization, matrix inversion, echelon
forms. Existence
and general form of solutions of linear systems. Minimum-length solutions*
of underdetermined systems. HW: 1.5(4, 5, 11, 19), 2.2 (3, 6, 8, 9, 10)
10 July: 2.1, 2.3, 2.4. Vector spaces, basis and
dimensions. Four fundamental subspaces.
11 July: 1.7, 2.5. Applications and special
matrices. Networks,
circuits, bridges and trusses*. Review
for exam 1.
12 July: More examples and
applications. Exam 1.
WEEKEND!
15 July: 3.1, 3.2. Orthogonality, cosines, angles. Projections onto and
orthogonal to specific vectors, projection matrices and their properties.
16 July: 3.3. Least-squares approximate solution of overdetermined
systems, normal equations, projections onto and orthogonal to subspaces. Analogy with minimum-length
solutions* of underdetermined systems.
17 July: 3.4, 3.1. Orthogonal and orthonormal matrices,
Gram-Schmidt orthonormalization, QR factorization and
algorithm. Orthogonality properties of
fundamental subspaces.
18 July: 4.1, 4.2, 4.3. Determinants, their properties and formulas. Review for Exam 2.
19 July: 4.4. Applications of
determinants. Exam 2.
WEEKEND!
22 July: 5.1. Eigenvalues and
eigenvectors, and their properties.
Characteristic polynomial. Symmetric matrices have real eigenvalues and orthogonal eigenvectors.
23 July: 5.2, 5.3. Diagonal form of a matrix.
Powers of matrix, other functions, Cayley-Hamilton
theorem, exponential of matrix.
24 July: 5.3, 5.4 Difference
equations, differential equations. Markov systems, population models. Asymptotic behavior for
long times.
25 July: 5.4. Mass-spring systems and higher-order DEs. Vibrational modes. Review for Exam 3.
26 July: More examples and
applications. Exam 3.
WEEKEND!
29 July: 6.1, 6.2. Quadratic forms.
Maxima, minima, saddle points. Tests for classification.
30 July: 6.3, 6.4. Generalized eigenvalue problems and Rayleigh
quotients. Applications.
31 July: Appendix A. Singular Value Decomposition.
1 August: Appendix A. SVD continued. Review for Exam 4.
2 August: Exam 4.
WEEKEND!
5 August: Course review.
6 August: Course review.
7 August: Final Exam.