MATH 410 SPRING 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.
14, 16 Jan: 1.1, 1.2, 1.4,
1.6. Review of vectors, matrices, linear systems. Geometric and algebraic
interpretations. Vector and matrix operations, transposes. *Norms of vectors and matrices.
21, 23 Jan: 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.
26, 28, 30 Jan: 2.1, 2.3,
2.4. Vector spaces, basis
and dimensions. Four fundamental subspaces.
2, 4 Feb: 1.7, 2.5. Applications and special
matrices. *Examples from math and
physics. Networks,
circuits, *equilibrium of mechanical structures.
6 FEBRUARY: EXAM 1 on chapters 1, 2.
9, 11, 13 Feb: 3.1, 3.2, 3.3.
Orthogonality, cosines, angles. Projections onto and
orthogonal to specific vectors, projection matrices and their properties.
Least-squares
approximate solution of overdetermined systems,
normal equations, projections onto and orthogonal to subspaces. *Analogy with minimum-length solutions of
underdetermined systems.
16, 18, 20 Feb: 3.4, 3.5. Orthogonal and orthonormal matrices, Gram-Schmidt orthonormalization,
QR factorization and algorithm. Orthogonality
properties of fundamental subspaces.
Discrete Fourier transform (skip FFT).
23, 25, 27 Feb: 4.1, 4.2, 4.3,
4.4. Determinants, their properties, formulas, and
applications.
1, 3 Mar: 5.1. Eigenvalues and
eigenvectors, and their properties.
Characteristic polynomial. Symmetric matrices have real eigenvalues and orthogonal eigenvectors.
5 MARCH: EXAM 2
on chapters 3, 4.
8, 10, 12 Mar: 5.2, 5.3, 5.4.
Diagonal form of a matrix. Powers of matrix, other
functions, Cayley-Hamilton theorem, exponential of
matrix (*complex integral formula). Difference
equations, differential equations.
13 – 21 MARCH SPRING BREAK!!!
22, 24, 26 Mar: 5.3, 5.4. Markov systems, population models. Asymptotic behavior for
long times. Mass-spring
systems and higher-order DEs. Vibrational modes, resonant frequencies.
29, 31 March, 2 April: 5.5,
Appendix A. Real
symmetric matrices and orthogonal diagonalizing
matrices. Singular
Value Decomposition and its properties and applications.
5 April: 6.1. Quadratic forms.
Maxima, minima, saddle points.
7 APRIL: EXAM 3
on chapter 5 and appendix A. (note this is Wednesday!)
9, 12, 14, 16 April: 6.2, 6.3,
6.4, 6.5. Tests for
classification of critical points of quadratic forms. Generalized
eigenvalue problems and Rayleigh
quotients. Applications,
e.g. finite-element method.
19, 21, 23 April: 7.1, 7.2,
7.3, 7.4 (NOT in detail!). Iterative
approaches to matrix computations. Connections between eigenvalues, norms,
convergence rates. *Gershgorin’s disk theorem for eigenvalues.
26, 28 April: Course review:
putting together comprehensive picture.
30 APRIL: EXAM 4 on chapters 6, 7 plus preceding material.
3, 5 May: Review for Final,
continue consolidating whole course.
MONDAY 10 MAY: FINAL EXAM