Math 2270 - Linear Algebra - Spring 2016

Projects

Complete a semester project by May 2. Suggestions appear below. You may use Maple or another computer program of your choice. The final project is expected to be in PDF format. Source code is expected in the format used to do the computation. For example, Maple code would be supplied in MW and MPL formats. Email is the preferred method for delivering the PDF and source code formats. A paper copy is expected two days before a project is presented in class. Presentations of the more interesting projects will be given in class during the last two lecture periods. 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 final 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 2.6. Was this economy productive?
Related: Section 2.6
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 Strang's 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: Section 4.8, Section 4.9
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 Sections 6.5, 6.6 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.

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: Chapter 7
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 1.9
Reference: Jennifer Kay, 2005 Computer Science document, http://elvis.rowan.edu/~kay/papers/kinematics.pdf,
Introduction to Homogeneous Transformations and Robot Kinematics

David Lay's Projects and Case Studies

The author of the 2270 textbook, 2016, has provided a number of interesting case studies in maple, mathematic and matlab sources, available at the Pearson web site, referenced in the introduction of the textbook. Here are the case study sources in maple, edited at Utah, to correct errors or add some details.
```case1-Linear-Models-in-Economics.mw
case2-Computer-Graphics-in-Automotive-Design.mw
case3-determinants-in-Analytic-Geometry.mw
case4-Space-Flight-and-Control-Systems.mw
case5-Dynamical-Systems-and-Spotted-Owls.mw
case6-Least-Squares-Solutions.mw
case7-Singular-Value-Decomposition-and-Image-Processing.mw
```
Find these edited files in this Directory

Previous Projects from 2270 Courses

There are a number of published projects from 2012, to show what is possible. Find the directory of files Here. The list of projects includes a number of distinct areas, some of which are richly supported in Lay's textbook. Embellishments to these existing projects would make a good project for 2016.
``` Presented Projects

Aldous, Arnold and Edwards: Waveforms and Spectrograms
Down, Firestone and Reed: Fractals: Iterated Function Systems and Linear Algebra
Edfrennes, Roddum and Thorsen: Cracking the Code: An Introduction to Hill Ciphers
Guckert: Fast Fourier Transform and the Modified Discrete Cosine Transform in MP3 Audio Compression
Kubly and Pellatt: Forecasting United States Real Gross Domestic Product
McGrath: Using Linear Algebra to Determine Spatial Autocorrelation: Geography
Weeks: The Vertex Adjacency Matrix: Illustrated Tales of (1) The Tortoise; (2) The Spanning Tree; (3) The Eel
Yizhou Ye: Image Compression by SVD and DCT

Submitted Projects and Incomplete Drafts

Azad and Wiser: Image Editing: Photos, RGB and Linear Algebra
Bess: Image Compression via DCT and SVD: A Matlab Investigation
Boyer: Gaussian Quadrature: An Application of Gram-Schmidt
Christensen: A Brief History of Linear Algebra
Gautam: Markov Chains and Nepal Voting Behavior
Koizumi: Sound Compression of WAV Files: Maple Investigation
Boya Li: Productive Economy: A Maple Investigation
Partridge: Fractals: A Maple Investigation
Wang and T. Ye: Relationship between Economic GDP and Mathematics
```

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.