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MATH 3333 Linear Algebra I

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Welcome to Math 3333: Linear Algebra I, Spring 2022!

Project Guide

Table of contents

Tentative due dates

(Official due dates are posted on Canvas > Assignments)

  • Wed, Feb 9, 2022, due on Gradescope: Preliminary exploration (brief explanation of a few topics you have read about) due. (You will have some time to read during class on Monday and ask me questions.)
  • Wed, Feb 16, 2022: Survey: forming teams due
  • Fri, Feb 18, 2022: During class, form teams based on topic interests (unless you already have a team). Each team should have 4 members. You may have time to start your ‘introduce your team’ slides.
  • Mon-Wed, Feb 21-23, 2022: During class, finalize your topic, create a team name, make a presentation video introducing yourselves as a team (due on Gradescope). This will serve as a warm-up to all the technology you will be using.
  • Fri, March 4, 2022, due on Gradescope: Check in 1 is due
  • Exam 1 is on Wed, March 9, 2022
  • Spring break March 14-18, 2022
  • Week of Wed, March 23, 2022: Work on your second draft
  • Week of Wed, March 30, 2022: Practice your presentation or finish your draft
  • Wed, April 13, 2022: During class, give live presentations of your presentation draft
  • Exam 2 is on Wed, April 20, 2022
  • Wed, April 29, 2022, due on Gradescope: video recording, writeup, and accompanying slides
  • April 29-May 4, 2022, due on Gradescope/YouTube: Answer questions from me or your peers who have watched the video.

Video: Introduce your team

The goal is to record a brief video where each person speaks for 1-2 min, introducing another person in the group. Discuss your group member’s professional, academic, other interests/activities.

Creating slides

  • You can create slides using …
  • Work on your slides

Recording

  • To record using Zoom …
    • Create a Zoom meeting. Share screen showing your slides.
    • Play with the settings so that everyone’s face is visible during the recording if possible.
    • Press the record button to record locally or record to the cloud. Recording will appear after you end the Zoom meeting.
  • You can also record using another tool that can record screen, e.g. QuickTime Player on a laptop or iPad/iPhone screen recording. If possible, play with the settings to find a way to make all team members’ face visible during the recording.

Sharing the recording and slides afterwards

  • Upload the video on YouTube with privacy setting: unlisted
    • Title of the video: All your initials [Team Name] Introduce your team video
    • Description: If you use online slides, create a shareable (read-only) link and put the link on the video’s description
  • Put the YouTube link and slides link on Gradescope: Video: Introduce your team. Only one person should submit on Gradescope for the entire team. Make sure to add all team members’ names after you submit.

Requirements

See requirements for slides and recording

Suggested Topics

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

Math concepts

  • diagonalizable matrices: You can 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 choose from

  • Come up with your own example
  • Examples from textbook Section 3.3
    • Example 3.3.1: Setting up bird population model
    • 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 textbook Section 3.3 (see solution manual):
    • Exercise 3.3.2(b),(d) more linear dynamical system
    • Exercise 3.3.31(b),(d) (about the bird population model from Example 3.3.1)
    • Exercise 3.3.34 (about the bird population model from Example 3.3.1)

2. Linear recurrence

Math concepts

  • diagonalizable matrices: You can learn it from the textbook pg 178-181 or any linear algebra resource
  • See the textbook Section 3.4. The math is similar to the above topic, linear dynamical systems

Some examples you can choose from

  • Come up with your own example
  • Example 3.4.1 number of ways to park cars and trucks (aka Fibonacci/Pingala numbers)
  • Example 3.4.2 Finding a formula for a linear recurrence using eigenvalues and eigenvectors
  • Example 3.4.3 Finding a formula

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

  • Exercise 3.4.1(b),(d) Finding a formula for a linear recurrence
  • Exercise 3.4.5 Counting words
  • Exercise 3.4.7 Counting ways to stack poker chips
  • 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

Math concepts

  • Read pg 322-326 of the textbook Section 5.7 (example of the data on sick days, doctor visits, and vitamin C)
  • (You do not need to include every definition mentioned in the book. Give just enough information to explain your examples)

Some examples you can choose from

  • Exercise 5.7.1
  • Exercise 5.7.2
  • Come up with your own example

4. Error Correcting Codes

Math concepts

Read the textbook Section 8.8

Some examples you can choose from

  • Come up with your own example
  • Examples from the book Section 8.8:
    • pg 473 Modular arithmetic (you can ignore 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)

Math concepts

Read the textbook pg 75 Directed Graph, the end of Sec 2.3 (an application of matrix multiplication to directed graph)

Some examples you can choose from

  • 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

Math concepts

This is an application of eigenvectors and matrix algebra

  • Read the textbook pg 128-131 Closed economy model
    • You don’t need to define every single definition used in the textbook. Skip definitions that you can do without!
  • Optional: See the textbook pg 131-132 Open sector model

Some examples you can choose from

  • Examples from the book Section 2.8:
    • 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)
  • Exercises from the book Section 2.8 (check the solution manual):
    • Section 2.8 Exercise 1(b,d); Exercise 2; Exercise 4
  • Come up with your own example

7. Vandermonde determinant and Polynomial interpolation

Math concepts

  • Polynomial interpolation Read the textbook pg 165-166
  • Vandermonde determinant Read the textbook pg 166-167

Some examples you can choose from

  • Do a computation of the general Vandermonde determinant (for the purpose of demonstration, use small 4x4 or 5x5 matrices)
  • Demonstrate using an example how the Vandermonde determinant plays a role in polynomial interpolation
  • Example 3.2.10 on textbook pg 165 (forestry)
  • Exercises from the textbook Section 3.2 (check the solution manual): Exercise 22; Exercise 23; Exercise 24
  • Research other uses of polynomial interpolation

8. 2D computer graphics

Math concepts

Read Section 4.5 pg 258-261 An application to 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.
  • You don’t need to define new vocabulary.

Special 2x2 matrices

  • Reflection with respect to the x-axis: Example 2.2.13
  • 90 degree counterclockwise rotation about the origin: Example 2.2.15
  • Other reflections and rotations: Sec 2.2 Exercise 11 (see solution manual or Google)
  • Vertical/horizontal expansions and compressions: Example 2.2.16
  • x-shear mapping: Example 2.2.17

9. An application to markov chains

Math concepts

  • Read the textbook Section 2.9 An application to Markov chains
  • Focus on the examples (only use the definitions you need to solve the examples!)

Some examples you can choose from

  • Example 2.91 about restaurant-eating: Application of matrix-vector multiplication
  • Another example from Sec 2.9
  • Exercises in Sec 2.9 whose solutions are explained in the student solution manual
  • Come up with your own example

10. Best approximation and least squares

Math concepts

  • Read the textbook Section 5.6 Best approximation and least squares
  • You do not need to include every definition mentioned in the book. Give just enough information to explain your examples)

Some examples you can choose from

Examples from the textbook Section 5.6

  • Example 5.6.1 (best approximation to a solution of a linear system)
  • Example 5.6.2 (best fitting function for average number of goals)
  • Example 5.6.3 (least squares approximating a line)
  • Example 5.6.4 (least squares approximating a quadratic polynomial)

Exercises from the book Section 5.6 (see solution manual):

  • Exercise 5.6.1 (best approximation to a solution of a linear system)
  • Exercise 5.6.2 (least squares approximating lines)
  • Exercise 5.6.3 (least squares approximating quadratic functions)
  • Exercise 5.6.7 (least squares approximating gravity)
  • Exercise 5.6.9 (best approximation for modeling wheat production)

11. LU factorization algorithm

This topic is recommended for students who are more interested in linear algebra theory than in applications

Math concepts

Read Section 2.7 in Chapter 2 of the textbook

Suggested presentation contents

  • Definitions of upper and lower triangular matrices
  • State the factorization theorem and maybe provide the idea behind the proof
  • Describe the algorithm for LU factorization
  • Demonstrate using examples
  • History of LU factorization

12. Other ideas

Check with me first before you start researching topics not listed above!

Pick a section of the textbook which is not covered in this class.

The class roughly covers the following section from the textbook:

  • Chapter 1 section 1, 2 Systems of linear equations
  • Chapter 2 section 1, 2, 3, 4, 6 Matrix algebra, inverses, linear transformation
  • Chapter 3 section 1, 2, 3 Determinants, eigenvectors
  • Chapter 4 section 1, 2 Vector geometry
  • Chapter 5 section 1, 2, 3, 4, 5 Subspaces, basis, dimension, eigenbasis for R^n
  • Chapter 6 section 1, 2, 3, 4 General vector spaces
  • Chapter 7 section 1, 2 Linear transformation for general vector spaces

Student roles

TBA

Structure of presentations

Slides

TBA

Video

TBA