Group portion of the final project

Prepare presentation materials (slides or other visuals) and a short group presentation similar to what you have been doing this semester. Create a recording. Give a live presentation and listen to other live presentations (on Wed Dec 16, 10:30am-12:30pm).

Time line:

Mon Dec 7 class: Decide on a tentative topic
Wed Dec 9 class: Prepare
Fri Dec 11 class: Prepare
Wed Dec 16 at 10:30am - 12:30pm: Give a live presentation during the final exam period for this class. (You can use this time to create a recording if you have not done so.)

Choose any of the topics which your group did not present on this semester:

1. Linear dynamical systems (bird population, diagonalizable matrices, dominant eigenvalue, graphical description)

diagonalizable matrices: Lecture 15c explains diagonalizable matrices. You can also learn it from the textbook pg 178-181 or any linear algebra resource.
dominant eigenvalue: See Theorem 3.3.7 for how to approximate steady state of a dynamical system using a dominant eigenvalue

Some examples you can use (choose just some of these or find your own):

pg 171, Example 3.3.1: Setting up bird population model
pg 182, Example 3.3.12: Solving the linear dynamical system of the bird population
pg 187-188 Graphical description of 2x2 linear dynamical system

Exercises from the book Section 3.3 (see solution manual):

pg 189 Exercise 3.3.2(b),(d) more linear dynamical system
pg 192 Exercise 3.3.31(b),(d) (about the bird population model from Example 3.3.1)
pg 192 Exercise 3.3.34 (about the bird population model from Example 3.3.1)

2. Linear recurrence (Textbook Section 3.4, very similar to linear dynamical systems)

diagonalizable matrices: Lecture 15c explains diagonalizable matrices. You can also learn it from the textbook pg 178-181 or any linear algebra resource.

Some examples you can use (choose just some of these or find your own):

pg 192 Example 3.4.1 number of ways to park cars and trucks (aka Fibonacci/Pingala numbers)
pg 194 Example 3.4.2 Finding a formula for a linear recurrence using eigenvalues and eigenvectors
pg 194-195 Example 3.4.3 Finding a formula

Exercises from the book Section 3.4 (see solution manual):

pg 196 Exercise 3.4.1(b),(d) Finding a formula for a linear recurrence
pg 197 Exercise 3.4.5 Counting words
pg 197 Exercise 3.4.7 Counting ways to stack poker chips
pg 197 Exercise 3.4.9 Modeling long-term wheat production
Warning: There are many myths on the role of Fibonacci/Pingala numbers in the sciences. Since it’s not easy to discern which publications on the internet can be trusted, it’s better to stick to just linear algebra examples in your presentation.

3. Correlation and Variance (Textbook Section 5.7)

(You do not need to include every definition mentioned in the book. Give just enough information to explain your examples)

pg 322-326 textbook reading (example of the data on sick days, doctor visits, and vitamin C)
pg 327 Exercise 5.7.1 and 5.7.2
Other examples

4. Error Correcting Codes (Textbook Section 8.8)

pg 473 Modular arithmetic (don’t worry about the definition of a field)
pg 475-476 Example 8.8.1 and 8.8.2 (example of arithmetic mod 2 and mod 17)
pg 476 Example 8.8.3 (matrix algebra for modular arithmetic)
pg 477-479 Error correcting codes explanation (you don’t need to explain every definition or theorem, only take what’s needed to understand the examples)
pg 480 Example 8.8.6 (linear codes)

Exercises from the book Section 8.8 (see solution manual):

pg 486 Exercise 8.8.1 (b),(d)
pg 486 Exercise 8.8.3(b) (inverse of 11 mod 19)
pg 486 Exercise 8.8.7(b) (Gaussian elimination with modular arithmetic)

5. Paths on a directed graph

Directed Graph Read the textbook pg 75, the end of Sec 2.3
Possible examples:
- Explain Exercise 2.3.26 on textbook pg 78. Check your answer with the solution manual, but give a more verbose explanation than what is given in the solution manual.
- We assume that a website does not link to itself, which means there shouldn’t a loop in the directed graph. Use the same directed graph as in Exercise 2.3.26, but remove the loop from v1 to v1. Answer the same questions but for this loopless directed graph.
- Make up a new directed graph (or find an already-drawn graph) and compute the number of paths of a certain length using matrix multiplication

6. Input-output economic models

Closed economy model Read the textbook pg 128-131 only
- You don’t need to define every single definition used in the textbook. Skip definitions that you can do without!
Possible examples:
- Example 2.8.1 (don’t simply take a screenshot of the example!)
- Example 2.8.2: Give names to the four industries. Explain the computation which is skipped by the texbook (you can use a software to row reduce and take a screenshot)
Some of these textbook exercises (check the solution manual):
- Section 2.8 Exercise 1(b,d); Exercise 2; Exercise 4
Optional: Open sector model See the textbook pg 131-132

7. Vandermonde determinant and Polynomial interpolation

Polynomial interpolation Read the textbook pg 165-166
Vandermonde determinant Read the textbook pg 166-167
Do a computation of the general Vandermonde determinant (for the purpose of demonstration, do a 4x4 or 5x5)
Discuss how the Vandermonde determinant plays a role in polynomial interpolation
Example 3.2.10 on textbook pg 165 (forestry)
Some of these textbook exercises (check the solution manual):
- Section 3.2 Exercise 22; Exercise 23; Exercise 24
- Research other uses of polynomial interpolation

8. 2D computer graphics

You don’t need theory or vocabulary you have not learned. Just use the facts that certain matrices and represent certain reflections/ rotations/ compression/ expansion/ shear mappings, and a product of certain matrices can have a desired effect on a 2D image.
An application to Computer Graphics Read Sec 4.5 pg 258-261
You don’t need to define new vocabulary.
Special 2x2 matrices:
- Reflection with respect to the x-axis: Example 2.2.13 pg 58
- 90 degree counterclockwise rotation about the origin: Example 2.2.15 pg 60
- Other reflections and rotations: Sec 2.2 Exercise 11 pg 63 (see solution manual or Google)
- Vertical/horizontal expansions and compressions: Example 2.2.16 pg 61
- x-shear mapping: Example 2.2.17 pg 61

9. An application to markov chains (Sec 2.9)

An Application to Markov Chains* Sec 2.9
Focus on the examples (only use the definitions you need to solve the examples)
Possible examples:
- Example 2.91 about restaurant-eating: Application of matrix-vector multiplication
- Another example in Sec 2.9
- Exercises in Sec 2.9 whose solutions are explained in the student solution manual

10. Other topic using linear algebra tools

Another topic (using basic linear algebra tools) which has not been a presentation topic this semester). Talk to me first before preparing the presentation.

Recording requirements of the group portion of the final

Between 5 and 10 minutes (it’s OK to go a little bit over if you have more than 4 people in your group)
The presentation should be appropriate for the students in a Math 3333 course. Assume that they have not seen your topic before but are familiar with all lectures.
Pin the speaker so that a thumbnail video of the speaker is visible
Introduce yourself (“Hello, I am …”) before you start speaking
(Optional) introduce your pets

Grading Scheme of the group portion of the final (same as past presentations)

The written portion will include partial credit, similar to the rubric for the worksheets.

Each item is worth 10%.

By Fri: Submitted (unfinished) link to work in progress and a pre-approved topic (due on Gradescope)
Final class meeting live session: Deliver a live presentation (even if you have already recorded a presentation). Turn your camera on while presenting.
Final class meeting live session: Listen attentively while another group is presenting (show your faces to the presenters). Chat questions (from both groups) should be saved and later submitted to Gradescope by the “Gradescope submitter” for each group.
Math: Explanation is at appropriate level for your audience (Math 3333 classmates who have never seen your topic but have seen class lectures or notes). Correct mathematics in presentation and slides.
Math: Terminology, facts, and examples are explained clearly.
Math: Effort has been made to do some independent thinking (not simply taking a screenshot of the source), coming up with your own examples.
Recording shows understanding of materials (not simply reading from a source).
Sufficient preparation and practice are evident in the recording. Video length meets the 5-10 minute length requirement (a longer video is OK for larger groups)
Visual aids (for example, slides or Jamboard) follow the requirements. They are legible and easy to see in the recording.
Speakers introduced themselves. It is easy to identify which speaker is speaking at a given time (use a thumbnail video or whole-screen video of presenters during recording). Voice is of appropriate volume and is clear.