Daily Assignments
December 3: Symmetric Matrices and Principal Component Analysis
 Mastery Quiz 12 due at midnight
 Read one of:
 You might also find one of these videos interesting. They don’t really go into the math behind it, but they talk a lot about when you might want to do this.
 This video by Luis Serrano is a really good explanation of PCA
 From 5:00 to about 12:15 he does an overview of the statistics concepts of mean, variance, and covariance. This isn’t really about the linear algebra, but you might find it useful anyway.
 From about 14:00 to 20:45 he does a quick overview of linear transformations, matrices, eigenvectors, and eigenvalues. You know this stuff already, but it also might be a really good review.
 At the 21 minute mark, notice how the eigenvectors automatically form an orthogonal basis!
 A good explanation of the purpose of PCA, from the perspective of data science and machine learning. Avoids a lot of the math technicalities. Focuses on how we can use a computer to do the computations for us. But has some great examples and visualizations.
 Here’s a more technical explanation of how PCA works
 He relates this all to something called Singular Value Decomposition, which is also cool, and we’ll talk about it next week.
 In step 4 at about 8:00, when he writes $CV = VD$, that’s just another way of writing the diagonalization $V^{1}CV = D$.
 This video by Luis Serrano is a really good explanation of PCA
December 1: Leastsquares solutions and linear regression
 WeBWorK due
 Read the solutions to Mastery Quiz 11
 Read one of:
 The notes §7.12
 LLM §6.56
 You may want to watch
 This video to show what a “least squares” solution really looks like.
 This video for another explanation of how projection solves regression problems.
November 24: Orthogonal Decomposition
 Read one of:
 The notes §6.4
 LLM §6.3
November 20: Mastery Quiz
 Mastery Quiz 11 due at midnight on Friday, November 20
November 19: Orthonormal bases
 Read one of:
 The notes §6.3
 LLM §6.2 and § 6.4
November 17: Inner Products
 WeBWorK due
 Read Solutions to mastery quiz 10
 Watch Essence of Linear Algebra, chapter 9: Dot Products and Duality
 Read one of:
 The notes §6.2
 LLM §6.7
 The rest of Beezer on Orthogonality
November 12: The Dot Product
 Mastery Quiz 10 due at midnight
 Read one of:
 The notes §6.1
 LLM §6.1
November 10: Diagonalization and Markov Chains
 Read solutions to mastery quiz 9
 Midterm Solutions are up
 Read one of
 The notes §5.4 and 5.5.
 LLM §5.3.
 The rest of Beezer on Simlarity and Diagonalization
November 5: Similarity and Trace
 Mastery Quiz 9 due at midnight
 Read solutions to mastery quiz 8
 Read one of
 The notes §5.23; you may also want to start §5.4.
 LLM second half of §5.2 “Similarity”; you may also want to start §5.3.
 The beginning of Beezer on Simlarity and Diagonalization; you may also want to look at the “Diagonalization” section.
November 3: WeBWorK due, no class
October 29: Change of Basis
 Mastery Quiz 8 due at midnight
 Watch (the rest of) the 3Blue1Brown video on change of basis
 Read one of
 The notes §5.1
 LLM § 4.7
 Back to Beezer on Change of Basis
October 27: Complex and Generalized Eigenvectors
 WeBWork due
 Read one of
 The notes § 4.34 (4.4 should be uploaded now).
 LLM §5.2 and §5.5
 Back to Beezer on Eigenvalues and Eigenvectors, the second half
 You may find it helpful to review Beezer on complex number operations
October 22: Determinants and the Characteristic Polynomial
 No mastery quiz this week
 Read one of
 The notes § 4.2
 LLM §3.12
 Beezer on Determinants and Properties of Determinants
October 20: No class; midterm due
 Midterm due midnight
 Practice Midterm
 Check out the solutions of mastery quiz 6.
October 15: Eigenvectors and Eigenvalues
 Mastery Quiz 6 due midnight on Thursday, October 15
 Watch Essence of Linear Algebra Chapter 14: Eigenvectors and Eigenvalues
 Read one of
 The notes § 4.1
 LLM §5.1
 Beezer on Eigenvalues and Eigenvectors but note he covers things in a different order
October 13: Isomorphisms
 WeBWork due
 Check the solutions of mastery quiz 5.
 Watch the first five minutes of Essence of Linear Algebra, Chapter 13: Change of Basis to solidify some of the stuff with coordinate systems. We’ll come back and handle the rest of this video in a few weeks. (We could do the other half now, but I think it’s a bit more compelling if we have defined eigenvectors first. Feel free to watch the whole video and see if you agree.)
 Read one of:
 The notes § 3.6
 Beezer on Invertible Linear Transformations
October 8: The Matrix of a Linear Transformation
 Mastery Quiz 5 due midnight
 You may want to rewatch Linear Transformations and Matrices as a warmup.
 Read
 The notes § 3.5
 Beezer on Matrix Representations
 LLM §5.4 on “The Matrix of a Linear Transformation” and “Linear Transformations from V to V” (pages 2913) (not terribly thorough)
October 6: Bases and Coordinates
 Check the solutions of mastery quiz 4.
 Read one of:
 The notes § 3.4
 LLM §4.4
 Beezer on Vector Representations (When Beezer writes $\mathbb{C}^n$ for the complex numbers, you can pretend he’s talking about $\mathbb{R}^n$ instead.)
October 1: Subspaces and Linear Transformations
 Mastery Quiz 4 due at midnight
 Read the notes § 3.23
September 29: Vector Spaces
 WeBWork due
 Take a look at the quiz 3 solutions.
 Watch Essence of linear algebra, chapter 15: Abstract Vector Spaces. (Don’t stress too much about his mentions of determinants and eigenvectors; we’ll get there.)
 Read one of:
 Notes § 3.1 * LLM §4.1 and skim §4.23
 Beezer on Vector Spaces, and look back at subspaces.
September 24: Bases, Dimension, and Rank
 Mastery Quiz 3 due at midnight
 Watch Inverse matrices, column space, and null space from Essence of Linear Algebra
 Read one of
September 22: Subspaces (and $LU$ factorizations)
 WeBWork due today
 Read one of
 Notes § 2.3
 LLM § 2.5
 Read one of
 Take a look at the solutions for quiz 2.
September 17: Matrix Inverses
September 15: Applications of Linear Functions

Mastery Quiz 2 Due at noon. Submit on Blackboard as one pdf file.

Read one of:
 Notes § 1.9. and §2.1
 LLM §1.10 and §2.1
September 14: Webwork due
September 10: Linear Transformations
 Watch Essence of Linear Algebra Chapter 3: Linear Transformations and Matrices
 Read one of
September 8: Solutions and independence of linear systems
 Mastery Quiz 1 Due at noon. Submit on Blackboard.

Watch Essence of Linear Algebra Chapter 2: Linear Combinations, Span, and Basis Vectors
 (We won’t talk about bases for a couple of weeks, but I actually think that this presentation is a more sensible way to think about what they’re doing. And there are some good visualizations of how to think about span and linear independence.)
 Read one of
September 3: Vectors and Matrix Equations:
 Read one of:
 LLM §1.31.4
 Beezer “Vector Operations”, “Linear Combinations”, “Spanning Sets”, “Matrix Operations”, “Matrix Multiplication”
 Online Notes §1.4.1.5.
September 1: Linear Equations and Row Echelon Form
 Please read the syllabus
 Claim your account on WeBWork (Username is your GWU email, password is GWID)
 Read the introduction in the course notes.
 Watch Essence of Linear Algebra, Chapter 1 by 3Blue1Brown
 Read at least one of the following:
 Sections 1.12 of Lay, Lay, McDonald
 Sections “What is Linear Algebra?”, “Solving Systems of Linear Equations”, and “Reduced RowEchelon Form” in Beezer
 Sections 1.13 of the online notes
 Optional/bonus: Watch this video on a technique called “Principal Component Analysis”. This is a sort of preview of what we want to be able to do by the very end of the course; it will mention a lot of ideas you haven’t been exposed to yet, but that you will see over the next few months.
Course Goals
This is a standard first course in linear algebra. The main topics are: linear equations; matrix algebra and equations; subspaces and bases; vector spaces; eigenvalues and eigenvectors; determinants; orthogonality and least squares. This corresponds to Chapters 1–7 of Lay, Lay, and McDonald.
By the end of the course, students will acquire the following skills and knowledge:
 Students will be able to find echelon forms of matrices, find a basis for the column space, row space and null space of that matrix, and determine if that matrix is invertible.
 Students will be able to determine if a set of vectors is linearly independent.
 Students will be able to calculate the eigenvalues and eigenvectors of matrices.
 Students will be able to diagonalize a matrix and use the diagonalization techniques to solve problems in other areas of mathematics.
The course syllabus is available here.
Course notes
Mastery Quizzes
The topics for the quizzes are:
 Systems of Linear Equations
 Vector Equations and Spans
 Linear Independence
 Linear Transformations
 Matrix Multiplication
 Matrix Inverses
 Subspaces
 Basis and Dimension
 Vector Spaces and Subspaces
 Vector Space Linear Transformations
 Bases and Coordinates
 The Matrix of a Linear Transformation
 Eigenvectors and Determinants
 Characteristic Polynomials and Finding Eigensystems
 Complex and Generalized Eigenvectors
 Change of Basis
 Similarity and Trace
 Diagonalization
 Dot Product and Projection
 Inner Products
 Orthogonal Decomposition
 Mastery Quiz 12 due at midnight on Thursday, December 3
 Mastery Quiz 11 due at midnight on Friday, November 20
 Mastery Quiz 10 due at midnight on Thursday, November 12
 Mastery Quiz 9 due midnight on Thursday, November 5
 Mastery Quiz 8 due midnight on Thursday, October 29
 Mastery Quiz 6 due midnight on Thursday, October 15
 Mastery Quiz 5 due midnight on Thursday, October 8
 Mastery Quiz 4 due midnight on Thursday, October 1
 Mastery Quiz 3 due midnight on Thursday, September 24.
 Mastery Quiz 2 Due at noon on Tuesday, September 15. Submit on Blackboard as one pdf file.
 Mastery Quiz 1 Due at noon. Submit on Blackboard.
Tests
 Midterm due midnight on Tuesday, October 20
 Final Exam
 [Practice Final]
 [Solutions]
Graphing calculators will not be allowed on tests. Scientific, nonprogrammable calculators will be allowed. I will have some to share, but not enough for everyone.
Textbook
The official textbook for Math 2184 is Linear Algebra and its Applications, Fifth Edition, by Lay, Lay, and McDonald (ISBN13: 9780321982384). I will be loosely following this book, and it will be very useful to have, but I will not be assigning problems out of it.
Another perfectly fine book is A First Course in Linear Algebra by Rob Beezer, which is available free online.