Linear Algebra Projects Spring 2012

Math 2270 - Linear Algebra - Spring 2012

Projects

Complete one of the following projects, due on the last day of class. Presentations of the more interesting projects will be given in class during the last two lecture periods, mixed with exam review. You may use Maple or another program of your choice. Work in groups of size one or larger, each group with a group leader, approved by the lecturer early in the semester. You may invent your own project or some variation of one of the suggestions, but please schedule an office visit before making a decision.

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
Use Matrix Data (csv)
Total Industry Output Vector (csv)
Sample Code (maple worksheet)
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
Compare the waveforms of several musical instruments playing the same note. Compare their energy spectra.
Sample Code (maple .mw)
puretone.wav
flute.wav
piano.wav
trumpet.wav

Statistics and Probability

Reconsider the height-weight data from Lab 3. Assume that each person underestimates their weight randomly by 2-4 percent. 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 feet 10 inches 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

Image Compression

Take a bitmap image (a digital photo) 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
Sample Code (maple .mw)
knot.bmp

Discrete Dynamical Systems

Compute orbits for some examples of discrete linear planar dynamical systems. Plot orbits for systems where the eigenvalues are real with absolute values less than one, equal to one, and greater than one. Plot orbits for systems whose eigenvalues are complex with modulus less than one, equal to one, and greater than one.

Consider the non-linear discrete planar 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}=1+ y_i - a (x_i)^2
y_{i+1}=b x_i
Do a few plots for a=1.4 and b=.3 and discuss the results.
What happens for different values of a and b?

Fractals

Create interesting fractals. See Professor Korevaar's fractal project page here
Also read this well-written 2005 master's thesis here by Petr Supina, titled Visualization of fractal sets in multi-dimensional spaces. Petr worked in mathematical applied information technology, within the Faculty of Nuclear Sciences and Physical Engineering.

Translations, Scaling, Rotations

Make a demonstration of computer graphics operations, to illustrate how to take a 3D image and display it in a different size, at a different location, rotated in 3D. Feel free to embellish this computer science and mechanical engineering project with your own ideas of what is interesting. Try to learn some elementary computer graphics, especially related to robotics, involving homogeneous coordinates, matrix operations, data organization and Object-Oriented programming.
Related: Section 8.7
Reference: Jennifer Kay, 2005 Computer Science document, http://elvis.rowan.edu/~kay/papers/kinematics.pdf,
Introduction to Homogeneous Transformations and Robot Kinematics

Audio Compression and MDCT

Make a 4-page paper and short presentation demonstrating the use of linear algebra in lossy audio compression, the modified discrete cosine transform (MDCT). Here's an undergraduate research project that may get you quickly into the topic, which has taken on more appeal since MP3 players have dominated the market, fueled by low-cost downloadable mp3 audio. AUDIO COMPRESSION USING MODIFIED DISCRETE COSINE TRANSFORM: THE MP3 CODING STANDARD http://www.mp3-tech.org/programmer/docs/jacaba_main.pdf
End of project suggestions.

Invented Projects

Other projects on different topics are encouraged. If you have an idea, then please discuss it in an office visit. Projects can be a group of just one, and once started, they can blossom into a group of two or more.

Please, don't hesitate to suggest an interesting topic. I left out medical topics, like the artificial heart research going on at Utah, mining applications, cloaking devices for the military, vision devices for the blind using ultrasound, solar wind research, solar panels, windmills, material science, chemical engineering, particle physics research, and an endless list of other possibilities.

Online Sample Projects

Finally, you can examine what others have done for linear algebra projects. A rich collection of slides, papers, matlab codes from 1997 to 2010 can be found here: