Schedule
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2-9: Discussion of objectives for Exam 1. Discussion of Exam 1, Fall 97.
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2-11: Exam 1.
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2-13: Discussion of the concepts of basis and dimension for vector spaces.
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2-16: Discussion of Exam 1. Discussion of vector and matrix norms.
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2-18: ConCept question on rank (item 90). ConCept question on subspace
(item 73). Computing a basis for a null space by hand and by MATLAB. End
of Vector Spaces outline.
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2-20: Begin Orthogonal Methods outline. We will look for the common thread
in three apparently dissimilar problems: finding the distance from a point
to a plane, modeling noisy data, and estimating the temperature profile
in a heated rod. Begin Project 2.
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2-23: Continued discussion of estimating the temperature profile in a heated
rod. Discussed Fourier's sine series orthogonal basis and the notion of
orthogonal and orthonormal bases in general. Passed back Project 1. Project
2 due March 6.
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2-25: Discussed orthogonal matrices, the minimum distance problem, and
Fourier's formula for computing the coefficients of the representation
of the closest vector given an orthogonal basis for the approximating subspace.
Orthogonal bases are a direct product of Fourier's method of separation
of variables for heat conduction problems. For linear modeling of noisy
data, we find an orthonormal basis using the singular value decomposition
(SVD). We briefly discussed the history of the orthogonal basis problem.
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2-27: Discussed the SVD and how it can be used to solve the minimum distance
problem.
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3-2: Discussed Problems 1-3 of Project 2.
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3-4: Finished discussing Project 2.
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3-6: Began the Eigenvalue Problems Outline. Discussed the geometry of the
matrix eigenvalue problem. Discussed Fourier's model of heat conduction
in a rod and the output from a simulation. Derived a Sturm-Liouvile problem
and its solution using 208 techniques. Looked at an approximating problem
which turns out to be a matrix eigenvalue problem. See the M-file eigen.m.
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3-9: We derived the characteristic equation associated with the eigenvalue
problem. We solved an eigenvalue problem for the symmetric matric A = [2
1 1; 1 2 1; 1 1 2]. We found the eigenvalues and their algebraic multiplicities
from the characteristic equation. We found a basis for each eigenspace
and then determined the geometric multiplicity of each eigenvalue. Finally,
we argued that A is nondefective and we found a diagonal matrix Lambda
and a nonsingular matrix X so that inv(X)*A*X = Lambda.
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3-11: Discussed theory and methods for finding the roots
of a polynomial equation. Discussed an inequality relating geometric and
algebraic multiplicity. Looked at the eigenvalue theory for a symmetric
matrix.
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3-13: Discussed the theory of first order systems in state
space form: x'(t) = A*x(t) where A is nxn. Compared to the theory learned
in 208 for linear, constant coefficient, homogeneous ordinary differential
equations.
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3-23: Review for Exam 2 on Friday March 27. Look at the objectives
for the following outlines: vector spaces, orthogonal methods, eigenvalue
problems.
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3-25: Review for Exam 2 on Friday March 27. Discuss Project
3 and team assignments.
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3-27: Exam 2.
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3-30: Beginning of Mathematical Modeling using PDE's outline.
The modeling equations for heat conduction in a rod.
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4-1: Discussion of Exam 2. Introduction to the method of
separation of variables.
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4-3: Teams should meet during the class period to set up
their schedules for Project 3. If possible go to the ECE 496 Bonus day
Friday, April 3 from 1:00-2:00 on the second floor of Riggs for pendubot
demos of stabilization in the nearly vertical position.
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4-6: More details on separation of variables. Some review
from 208.
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4-8: The theory of generalized Fourier series. MATLAB simulations
with fsex1.m.
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4-10: More MATLAB simulations to explore convergence rates
for generalized Fourier series. What determined the number of terms required
to obtain a good approximation?
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4-13: Sturm-Liouville problems for jigsaw homework.
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4-15: Sturm-Liouville problems for jigsaw homework.
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4-17: Sturm-Liouville problems for jigsaw homework. Plan
to attend the ECE 496 Competition Day, Friday April 17 from 1:00-2:00 in
front of Riggs for pendubot demos of swing-up from the down position to
a stabilized vertical position. This should be interesting. Third exam
will be posted.
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4-20: Sturm-Liouville problems for jigsaw homework.
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4-22: Due date for take home third exam and jigsaw homework.
Course evaluation. Workgroup evaluation. Discussion of MATLAB simulations.
Pass back Project 3.
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4-24: Pass back third exam and jigsaw homework. This is will
be a review day for the final. There is no reading day this semester. This
is a day of classes.
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4-27 to 5-1: Office Hours April 27-May1: 10:00-11:00.
Other times by appointment.
Topical Outlines
Please bring a copy of the topical outlines to class. Study the appropriate
outline before coming to class and be prepared to answer questions about
it. During lectures you may find it convenient to annotate these outlines.
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Solving Linear Systems by Gauss Elimination.
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Elementary Matrices and Matrix Factorization.
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Vector Spaces.
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Orthogonal Methods.
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Eigenvalue Problems.
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Mathematical Modeling using PDE's.
Course Objectives
By the end of the class discussion of each outline, you will be able to
...
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Solving Linear Systems by Gauss Elimination.
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Elementary Matrices and Matrix Factorization.
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Vector Spaces.
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Orthogonal Methods.
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Eigenvalue Problems.
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Mathematical Modeling using PDE's.