# Math 2270-003 - Linear Algebra - Fall 2010

# Projects

Complete one of the following projects, due on the last day of class.
You may use Maple or another program of your choice. You must work
independently.
You may also come up with your own project or do some variation of one
of the suggestions, but you should consult me first.
## Economics

Use the 2008 Summary Use Annual I-O Table found at
http://www.bea.gov/industry/io\_annual.htm
to construct a consumption matrix as in Section 8.3. Was this economy productive?

Related: Section 8.3

Note that the Use Matrix is not the same as the consumption matrix in
the book. To get the consumption matrix you must rescale column j of
the Use Matrix by dividing by entry j of the Total Industry Output Vector.
## Music

Compare the waveforms of several musical instruments playing the same
note. Compare their energy spectra. Comment on how the energy spectrum
looks in relation to the sound the instrument makes.

Related: Lab 4, Section 8.5, Section 7.3, Section 10.3

## Statistics and Probability

Reconsider the height-weight data from Lab 3. Assume that each person
underestimates their weight randomly by 2-4\%. Use the weighted least
squares method of Section 8.6 to find a more accurate model function
for the height-weight data. Plot the data, new model, and old model
together on the same set of axes. Pick a height (it was 5'10'' in Lab
3) and compute the expected weight of a person of that height using
the two different models.

Related: Lab 3, Section 8.6

## Computer Graphics

Take a 3--dimensional wireframe model and move it around using the
techniques of Section 8.7.

Related: Section 8.7

## Image Compression

Take a bitmap image and compress it using two different methods, using
the largest singular values of the SVD and using the largest
values of the Discrete Cosine Transform. Experiment with how many values you
must retain to have acceptable image quality. Calculate the
compression ratio of your image. Show pictures of some
basis vectors of the DCT encoding.

Related: Section 6.7, Section 7.2, Section 10.3

## Discrete Dynamical Systems

We have talked a lot about discrete linear dynamical systems.
Compute orbits for some discrete linear dynamical systems in the
plane. Plot orbits for systems where the eigenvalues are real with
absolute values less than one, one, and greater than one. Plot orbits
for systems whose eigenvalues are complex with norm less than one,
one, and greater than one.

Consider
the non-linear discrete dynamical system that takes a point
(x_i,y_i) in the plane and moves it to the point (x_{i+1},y_{i+1})
where:

x_{i+1}=y_i+1-ax_i^2

y_{i+1}=bx_i

Try setting a=1.4 and b=.3. Plot some orbits and discuss the
results.

What happens for different values of a and b?

## Fractals

Make some fractals. See Part B of http://www.math.utah.edu/~korevaar/2270fall09/mapleproj1.pdf
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Last updated August 23, 2010.

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