Research Sessions

Goals and Objectives: You may work alone or with one other person. You will research a topic related to the course that you are interested in and will communicate your expertise. The presentations are modeled after what happens at research conferences, Appalachian's student research day, and science fairs. Your project will be graded based on the linear algebra connections, the clarity, quality and creativity of the following:
  1. Part 1: A review of any class work that relates to your topic in Part 2 [3-sides typed (single-spaced), including images]
    Include the relevant definitions, mathematical symbols and notation, pictures, theorems, examples, algebraic, geometric, and numerical representations from class, homework and tests that relate to your topic. If you have connections to concepts that we saw very regularly, such as a concept like Gaussian elimination, then bullet point lists of the places we used this in class would help summarize what we covered in class. To create this review, use your class notes as well as our online resources. This will require you to dig for as many connections as possible, giving a creative way of reviewing.

  2. Part 2: An extension of class work that you create related to your topic and linear algebra. This could be covering new material, making connections to your field, looking up history, seeing how linear algebra is programmed... The type and format are your choice - here are some ideas, for example:
    1. the beginnings of a more extensive research project
    2. a summary of what you have learned after researching a topic--- in your own words [it could be paragraph or bullet point format and it could be longer, but 1 or 2 pages should be sufficient in many cases]
    3. a computer program you work on and report back on how that went---what was already available to you (or not) in the programming language you choose, what you tried, and what you would do with more time
    4. a summary of linear algebra connections to a different course you have already taken or will take
    5. a demo you create
    6. a representation of historical information that you create
    7. a classroom worksheet that you create as you research and report back on classroom standards related to linear algebra
    8. an experiment that is connected to linear algebra and report back on how that went
    The only requirement for this portion is that you create something in your own words that extends/connects to our course content in some way. There are lots of possibilities here - I encourage you to be creative!

  3. An annotated reference list (to turn in). The annotations are brief comments about how you used each reference in your project. Most project should have some scholarly sources. Faculty, past classes and experiences can also be listed as references. Be sure to acknowledge the source citations of pictures but there is no need to annotate them.

  4. Research session presentations and peer review. Bring a printed version of all of your work. We will divide up the class into two sessions (half the class will stand next to their work as the other half examines the projects, and then we will switch roles). If you work with another person, they will be in the other session so you should be prepared to present the entire project. During your session, you must stand by your work to discuss your topic and answer questions. The presentation component typically involves a group of 1 or 2 students at a time listening to and looking at your project so they can take notes for peer review.
All components must be typed products (except the peer review) that you create yourself in your own words, and that look professional and flow well. Mathematics symbols and notation should be typed in a program like LaTex or Maple.

Here is a rubric for the final project

Here are some sample projects:
Dark Matter Accretion and the Hessian Matrix by Collin Sweeney [could be improved by removing Appendix A and just making the examples a full part of the in-class connections]
Finding Determinants Through Programming by Wyatt Andresen [would benefit by including pictures in part 1 and fixing typos]

Many past students have used a word processor (sometimes with Maple output pasted in) or LaTeX. Here is Wyatt's LaTeX file that you can place this in a real-time editor like Overleaf).

Note that part 1 should be purely class review. Any new material belongs in part 2. For instance, say your extension incorporates determinants in some way. Then the new extension is in part 2, while a review of what we already did related to determinants belongs in part 1. Some past students reported that they have found it helpful to think of part 1 as a review of class notes and hw as if they were studying for a final exam [without the exam component - instead the product is finding the connections].

This project connects in a variety of ways to the four general education goals for all students at ASU:
  • Thinking Critically & Creatively [research and creative product]
  • Communicating Effectively [writing, speaking and reflecting]
  • Making Local to Global Connections [how our class work applies in many other settings, multiple perspectives]
  • Understanding Responsibilities of Community Membership [citations, peer review, actively listening to each other's perspectives and presentations...]

    Sample Project Ideas I encourage you to be creative and find a topic that relates to linear algebra and interests you!

    Because we have so many intended computer science majors, I have designated topics that could easily connect to cs via a *.

  • Anomaly detection and linear algebra *
  • Applications of higher dimensional vector spaces to computer learning in order to diagnose heart disease, breast cancer, and use sonar signals to distinguish rocks from mines *
  • Collision detection and linear algebra *
  • Connections between linear algebra and calculus III, a physics, computer science, geology, or other class you have already taken, or with research experiences or your field *
  • Covariance matrix and mean vector *
  • Determinants and the eight queens problem *
  • Eigenfaces *
  • Fit points to a line or plane *
  • Frustum culling and linear algebra *
  • Gershgorin circle theorem and applications to flutter of an aircraft
  • Golden mean and matrices
  • Google and linear algebra *
  • How Does the NFL use Linear Algebra to Rate the Passing Ability of Quarterbacks?
  • History of a topic in linear algebra [like Gaussian elimination, linear transformations, vectors...]
  • Hill cipher and eigenvalues
  • Image alignment and linear algebra *
  • Least squares sports ratings *
  • Lorentz transformation and special relativity
  • Markov chains and actuarial sciences
  • Neural networks and linear algebra *
  • Optimal sustainable harvesting and linear algebra
  • Orthogonal matrices and Gram Schmidt *
  • Point on which side of line, plane, hyperplane? *
  • Principal component analysis and linear algebra in machine learning or image processing *
  • A proof we did not cover in class, like "Any orthogonal set of n nonzero vectors in Rn must be a basis for Rn"
  • A part of the book we did not cover in class, like Cramer's rule *
  • Quantum mechanics and eigenvalues
  • Recommendation engines and linear algebra *
  • Rotation matrices, gimbal lock, and the space shuttle
  • Statistics and linear regression and linear algebra
  • Singular value decomposition in image compression *
  • Visualization and linear algebra *
  • Many many other possibilities - I am happy to help in office hours!