## Math 2270 - Linear Algebra - Spring 2018

### Projects

Complete a semester project and submit it 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 print of the PDF is not expected, although classroom copies and/or an internet link is recommended, if your group has a presentation. 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 consult with Dr. G 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

./pdf-documents/fractals-petr-supina-2005-bp_vis.pdf 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. Files with extension PDF require a portable document format reader like Adobe Reader. Files with extension .mw or .mws require Maple version 18 or later to read the worksheet.
```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

Projects from 2270 in 2012, Strang's Textbook. 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 2017. Files with extension PDF require a portable document format reader like Adobe Reader. Files with extension .mw or .mws require Maple version 18 or later to read the worksheet.
```Presented Projects 2012 Directory Link 2012

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
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 2012 Directory Link 2012

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
```
Projects from 2270 in 2016, Lay's Textbook. Below find projects in PDF form with computer code in most cases. Some of these projects are first quality. Others are definitely incomplete. Files with extension PDF require a portable document format reader like Adobe Reader. Files with extension .mw or .mws require Maple version 18 or later to read the worksheet.
```Directory of 2270 Projects Spring 2016

Presented Projects 2016

Tyler Adams, David Moody, Haysun Choi:  Cryptography and the Enigma Machine

Veronica Dean-Perry and Marie Novozenya: Least squares fitting of weight data
deanPerryVeronica-novozenyaMarie-least-squares.pdf

David Ethan Hamilton, Daniel Merrell, Aaron Kramer, Marty Simmons, and Carlos Guerra: Economics Stochastic Models
Directory hamiltonEthan

Daniel Mattheiss:  Fractals generated by iteration of affine mappings
Directory mattheissDaniel

Barrett Williams: Predicting Financial Security Prices using the FFT
Directory williamsBarrett

Jie Zhang and Pratusha Bopanna: Markov Chains and Music Composition
Directory zhang-bopanna

Submitted Projects 2016

Christian Butler: Inheritance and Population Genetics
butler-Christian-inheritance-population-genetics.pdf

John Chambers: Spectral differences of sound and harmonics using Fourier Transformations
chambersJohn-Spectral-Differences-Sounds-and-Harmonics.pdf

William Garnes: Fractal generation using linear and affine transformations
Directory garnesWilliam-fractals

Benson Haglund
Leontief Consumption Matrices and Economics Models
Directory haglundBenson

Conner Kuhn
A Comparison of SVD and DCT Image Compression Methods
Directory kuhnConnor

Nicholas Lloyd
OpenGL Matrix/Vector Manipulation using SFML: Applications of Linear and Affine Transformations
lloydNick-open-GL-computer-graphics.pdf

Jessica Murdock
Image Manipulation Using Mathematica: Matrix Representations of Images and Linear Transformations
murdockJessica-Image-Manipulation-Mathematica.pdf

Nathan Romriell: Fractals, Julia Sets and Mandlebrot Sets: Linear and Affine Transformations
Directory romriellNathan

Dario Sanchez: Balancing Chemical Equations using Systems of Linear Algebraic Equations
sanchezDario-chemistry-balancing-chemical-equations.pdf

Christopher Sannar: Representation of Musical Scores by Vectors and Matrices
sannarChristopher-Music-scores-and-matrices.pdf

Orenda Williams: Manipulation of Digital Images using Matrices
williamsOrenda-matrix-operations-digital-images.pdf

Incomplete Projects 2016

Mahalia Lotz:  Spectral properties of music
LotzMahalia-music-spectral-properties-fourier-analysis.pdf

Jiwon Nam: Image Compression
namJiwon-image-compression.pdf

Seth Reelitz: Hill Cipher
reelitzSeth-hill-cipher.pdf

```
Projects from 2270 in 2017, David Lay's Textbook. Below find projects in PDF form with computer code in most cases. Some of these projects are first quality. Others are definitely incomplete. In 2018, please choose a project topic that is not too broad and not too difficult. Files with extension PDF require a portable document format reader like Adobe Reader. Files with extension .mw or .mws require Maple version 18 or later to read the worksheet.

Directory of 2270 Projects Spring 2017

### Presented Projects

1. Joseph Pugliano and Brandon Sehestedt
Cryptography: Matrices and Encryption Directory
2. Alex Hawks and Matthew Westberg
Cracking the Hill-Cipher: how to break an encrypted message Directory
Image Compression with Haar Wavelets Directory
4. Mark Van der Merwe, Ann Wilcox, Andrew Haas
Image Compression using Singular Value Decomposition and Discrete Cosine Transformation Directory
5. Jeremy Jakobs and Nathan Rogers
Linear Algebra and its Applications in Genetics Directory
6. Jasper Slaff
Using Homogeneous Coordinate Systems in Computer Graphics Directory
7. Miriam Galecki
Principle Component Analysis: Barrier Islands Example Directory

### Submitted Projects

PNG vs JPEG Compression Directory
2. Asher Sorensen
Photograph Manipulation Through Kernel Convolutions Directory
3. Brennan Benavidez
Instrument Wavelength Comparisons: Music and Markov Chains Directory
4. Conner Schacherer
DCT Compression Directory
5. Chris Billingsley
Fractals: Starting From The Base Directory
6. Dylan Johnson
Graph Theory and Linear Algebra Directory
7. Joe Narus
Image Compression and Quality Directory
8. Jonah Barber
Discrete Dynamic Systems That Approximate Motion Through Velocity and Acceleration Vector Fields Directory
9. Joshua Rosen and Alexandra Bertagnolli
Using Markov Chains to Procedurally Generate Text Directory
10. Josh Ulrich
Statistics and Probability: Estimating Body Weight with Weighted Least Squares Method Directory
11. Mark Lavelle
An Application of Linear Algebra in Least-Squares Solutions and Statistical Inference Directory
12. Patrick Ekel and Austin Purdie
Using Ordinary Least Squares to Predict National GDP Directory
13. Rahul Ramkumar
Convolutional Neural Networks to classify images of handwritten text Directory
14. Julia Vonessen, Gabrielle Legaspi and Saleema Qazi
Hill Ciphers: An Application of Linear Algebra Directory
15. Gabriella Barnes and Veronika Gribenko
Productivity of the 2008 Economy Directory
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.