• Course Information
  • Introduction
  • Citation
  • In Development
  • Errata
  • Consulting/Speaking
  • Course Syllabus
  • R Resources
  • Course Details
• Course Units
  • Getting Started
  • R Bootcamp
  • How Computers Store Numbers
  • Fundamentals of Linear Systems
  • Solving Linear Systems
  • Recap Exercise I
  • Interpolation
  • Least Squares I
  • Least Squares II
  • Eigenvalues I
  • Eigenvalues II
  • Principal Component Analysis
  • Recap Exercise II

Course Information

Introduction

I am Prof. Chad Topaz. Welcome to the public facing version of my Computational Linear Algebra course. Since this site is for public sharing, I have eliminated certain items related to surveying my students, performing course evaluation, student assessment, and so forth.

I enthusiastically credit colleagues from my time at Macalester College: Tom Halverson, Danny Kaplan, David Shuman, and Lori Ziegelmeier. Their versions of a Computational Linear Algebra course have influenced my own course, and I’m grateful to them for sharing wisdom and materials.

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Citation

If you use this site as a student (for your own learning) or as an instructor (for teaching your own course) I would appreciate you letting me know so that I can track impact of the work I have done to create it. If you do adopt materials from this site, please make sure to credit me.

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In Development

This course site is perpetually in development. I regularly revisit my course materials in order to improve them.

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Errata

If you find any problems with course materials (a typo, code that doesn’t work, a broken link – anything at all) please let me know by contacting me here.

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Consulting/Speaking

I have built this public facing site as a free, shareable public good. If you are interested in assistance with technical/mathematical aspects of this course or related consulting, please contact me here. If you would like me to speak to your department/institution/organization about curriculum and/or pedagogy (of this course, or of anything else) then please see my Speaking page and contact me here.

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Course Syllabus

Rather than using a traditional text-based course syllabus, I use a rather concise graphical syllabus; see below. I designed this syllabus using Piktochart. A similar tool you could consider is Canva. Some details of this syllabus are not relevant for a broad public audience, but I include the full syllabus nonetheless in order to give a flavor of the course and the philosophy I bring to it.

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R Resources

This course uses the R programming language in the RStudio integrated development environment. RStudio Cloud is a free version of RStudio that you can run on any device that has an internet connection and a web browser.

Here are some help resources for R.

• Learning R

  • DataCamp [See Getting Started below for a discussion of DataCamp’s problematic past; it is imperative to read this in order to make an informed decision about usage]
  • DataQuest
  • Learn R in Y Minutes
  • RStudio Primers
  • R for Data Science

• Cheat Sheets

  • RStudio Environment
  • Base R
  • R Markdown
  • Data Visualization

• Help Resources

  • Stack Overflow
  • My Favorite Choice

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Course Details

A few characteristics of my on-the-ground course include:

• Mastery-based assessment
• Flipped classroom style
• Introductory activities that get students oriented to the course
• Organization around weekly blocks of material
• Substantial in-class small-group discussion
• Positive psychology

To enable you to make easy and flexible use of the resources here, I have eliminated most of these elements from this public facing version of the course. Essentially, what I am providing here is a core dump of materials I use. The materials are linked throughout the various course units. Additionally, you can access the raw materials in this respository’s subdirectories:

• /coursebook
• /demos
• /activities
• /psets
• /recap
• /worksheets

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Course Units

Getting Started

• Videos

  • Why is Linear Algebra Useful (9:56)

• Reading

  • The Mathematics of Mass Testing for COVID-19
  • DataCamp Hid a Sexual Harassment Incident by its CEO [CW/TW: Sexual Harassment]. DataCamp is this course’s default resource for learning R because it is the best-designed and most effective resource I know of. However, DataCamp has a problematic past, and I feel an ethical imperative to be transparent about this, which is why I am having you read the article linked above. If you feel you cannot or should not use DataCamp, I encourage you use an alternative option, including one of the choices I have listed in the R Resources section. In case it impacts your decision, please know that DataCamp does not gain financially from us using it, as instructors at colleges/universities can get six free months of DataCamp for their students. In any case, you are 100% free to make whatever choice you want and I will support you fully!

• DataCamp Coding Practice

  • Introduction to R (course)

R Bootcamp

• Videos

  • Horner’s Method (17:16) 
  • Intro to Taylor Series (12:42)
  • Why Taylor Series Actually Work (9:34)

• Reading

  • Course Book: R Bootcamp
  • Machine Learning Has Been Used to Automatically Translate Long-Lost Languages
  • The Best Bits

• DataCamp Coding Practice

  • Getting Started with R Markdown (chapter)
  • Introduction to Linear Algbra (chapter)
  • Conditionals and Control Flow (chapter)
  • Loops (chapter)
  • Functions (chapter)

• Activity - R Bootcamp [.md for viewing] [.Rmd for editing]

• Pset - R Bootcamp [.md for viewing] [.Rmd for editing]

• In-class code examples

• Challenge

  • Many students take the formula for Taylor series as a given, but it’s valuable to understand how it comes from calculus. Derive the Taylor series formula [hint: start with the Fundamental Theorem of Calculus].

• Looking ahead

  • 8.2 - 7.2 - 1 = ?
  • 10¹⁸ + 100 − 10¹⁸ = ?

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How Computers Store Numbers

• Videos

  • Floating Point Numbers (17:29)
  • Floating Point Representation and Rounding Error (16:26)

• Reading

  • Course Book: How Computers Store Numbers

• Activity - How Computers Store Numbers [.md for viewing] [.Rmd for editing]

• Pset - How Computers Store Numbers [.md for viewing] [.Rmd for editing]

• In-class code examples

• Challenge

  • TBD

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Fundamentals of Linear Systems

• Videos

  • Two Views of Linear Equations (18:21)

  • The Big Theorem (4:35)

  • Gaussian Elimination (7:11)

  • Matrix Norms (9:56)

  • Condition Number [starting at 8:30] (7:54)

• Reading

  • An Ill-Advised Linear Algebra Tutorial

  • Course Book: Fundamentals of Linear Systems

• Activity - Fundamentals of Linear Systems [.md for viewing] [.Rmd for editing]

• Pset - Fundamentals of Linear Systems [.md for viewing] [.Rmd for editing]

• Challenge

  • A theory problem: prove that the condition number of a matrix is equal to the norm of the matrix times the norm of the inverse of the matrix.

  • An applied problem: do the “reality check” on the Euler-Bernoulli beam in section 2.4 of Sauer’s Numerical Analysis.

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Solving Linear Systems

• Videos

  • LU Factorization (8:03)

  • Iterative Methods for Ax = b [first 9:09 only] (9:09)

• Reading

  • Course Book: Solving Linear Systems

• Activity - Solving Linear Systems [.md for viewing] [.Rmd for editing]

• Pset - Solving Linear Systems [.md for viewing] [.Rmd for editing]

• Other Resources

  • In-class code examples

• Challenge

  • Learn about the secant method for root finding and calculate (analytically) the order of convergence. Check your result with a numerical example.

  • Read and understand the original paper on Katz centrality.

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Recap Exercise I

• Recap #1 [.md for viewing] [.Rmd for editing]

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Interpolation

• Videos

  • Introduction to Polynomial Interpolation (13:46)

  • Lagrange Interpolating Polynomial (8:10)

  • Chebyshev Polynomials (up to 9:27)

  • Polynomial Fitting and Splines (6:51)

• Reading

  • Course Book: Interpolation

• Activity - Interpolation [.md for viewing] [.Rmd for editing]

• Pset - Inteprolation [.md for viewing] [.Rmd for editing]

• Other Resources

  • In-class code examples

• Challenge

  • Learn how to implement interpolation via Newton divided differences and implement the technique in R.

  • Prove the formula for Chebyshev interpolation nodes.

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Least Squares I

• Videos

  • Vector Projections

  • Least Squares I: Matrix Problems (up to 8:32)

• Reading

  • Course Book: Least Squares I

• Activity - Least Squares I [.md for viewing] [.Rmd for editing]

• Pset - Least Squares I [.md for viewing] [.Rmd for editing]

• Other Resources

  • In-class code examples

• Challenge

  • TBD

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Least Squares II

• Videos

  • Graham-Schmidt Orthogonalization (6:37)

  • QR Factorization (8:31)

  • An Example of QR Decomposition (12:24)

• Reading

  • Course Book: Least Squares II

• Activity - Least Squares II [.md for viewing] [.Rmd for editing]

• Pset - Least Squares II [.md for viewing] [.Rmd for editing]

• Other Resources

  • In-class code examples

• Challenge

  • TBD

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Eigenvalues I

• Videos

  • What Eigenvalues and Eigenvectors Mean Geometrically (9:08)

  • Using Determinants to Compute Eigenvalues and Eigenvectors (4:35)

  • Example: Computing Eigenvalues and Eigenvectors (7:32)

  • Diagonal Matrices are Freaking Awesome (6:27)

  • How the Diagonalization Process Works (7:29)

  • Compute Large Powers of a Matrix via Diagonalization (3:00)

  • Full Example: Diagonalizing a Matrix (10:07)

  • Power Method for Eigenvalues (8:19)

  • The Power Method (11:59)

  • Introduction to PageRank (8:44)

• Reading

  • Course Book: Eigenvalues I

  • Link Analysis and Web Search, pp. 397 - 417

• Activity - Eigenvalues I [.md for viewing] [.Rmd for editing]

• Pset - Eigenvalues I [.md for viewing] [.Rmd for editing]

• Other Resources

  • In-class code examples

• Challenge

  • TBD

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Eigenvalues II

• Videos

  • Intro to Markov Chains and Transition Diagrams (11:24)

  • Markov Chains and Transition Matrices (6:53)

  • Absorbing Markov Chains Definition (3:35)

  • Absorbing Markov Chain Standard Form [first 2:08 only] (2:08)

• Reading

  • Markov Chains Explained Visually

• Activity - Eigenvalues II [.md for viewing] [.Rmd for editing]

• Pset - Eigenvalues II [.md for viewing] [.Rmd for editing]

• Other Resources

  • In-class code examples

• Challenge

  • TBD

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Principal Component Analysis

• Videos

  • Principal Component Analysis (PCA), Step-by-Step (21:57)

  • Principal Component Analysis (PCA) (26:33)

  • PCA in R (8:56)

• Reading

  • Principal Component Analysis: Resources for an Essential Application of Linear Algebra

• Activity - Principal Component Analysis [.md for viewing] [.Rmd for editing]

• Pset - Principal Component Analysis [.md for viewing] [.Rmd for editing]

• Other Resources

  • In-class code examples

• Challenge

  • Find a high-dimensional data set that interests you. Perform PCA on it to see if there is a lower-dimensional structure that captures much of the variation in the data. How many principal components should you keep to be faithful to your particular data set? Do you see a clear interpretation of the loadings? Does your lower dimensional representation of the data reveal anything about its structure?

  • Discuss some limitations of PCA. Is it possible for data to have a low-dimensional structure that is not well-captured by PCA?

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Recap Exercise II

• Recap #2 [.md for viewing] [.Rmd for editing]

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