Math
2270 - Linear Algebra - Spring 2017
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. 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
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 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. In 2017, 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 2016
Presented Projects 2016
Tyler Adams, David Moody, Haysun Choi: Cryptography and the Enigma Machine
adamsTyler-moodyDavid-choiHaysun-CryptographyTheEnigmaMachine.pdf
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
Kyle Heaton and Braden Scothern:
scothernBraden-heatonKyle-hamming-codes.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
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.