# Math 2184: Linear Algebra I Section 10 Fall 2020

#### Contact Info Fall 2020

Office: Blackboard
Email: jaydaigle@gwu.edu

Office Hours:

• Monday 11 AM -12 noon
• Wednesday 5-6 PM
• Thursday 3-4 PM

#### Course Information

Lecture:

• TR 11:10 AM–12:25 PM
• on Blackboard

Textbook:

## Daily Assignments

#### December 3: Symmetric Matrices and Principal Component Analysis

• Mastery Quiz 12 due at midnight
• The notes §7.3
• LLM §7.1 and §7.5 (But note that the explanation in §7.5 is really bad.)
• I actually found this pdf really helpful.
• 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 $V^{-1}CV = D$.

#### December 1: Least-squares solutions and linear regression

Slides

• WeBWorK due
• Read the solutions to Mastery Quiz 11
• LLM §6.5-6
• 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

Slides

• The notes §6.4
• LLM §6.3

#### November 19: Orthonormal bases

Slides

• The notes §6.3
• LLM §6.2 and § 6.4

Slides

Slides

Slides

#### November 5: Similarity and Trace

Slides

• Mastery Quiz 9 due at midnight
• Read solutions to mastery quiz 8
• The notes §5.2-3; 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.

Slides

Slides

Slides

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#### October 13: Isomorphisms

Slides

• 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.)

Slides

#### October 6: Bases and Coordinates

Slides

• Check the solutions of mastery quiz 4.
• 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.)

Slides

Slides

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#### September 22: Subspaces (and $LU$ factorizations)

Slides

• WeBWork due today
• Notes § 2.3
• LLM § 2.5
• Notes § 2.4
• LLM § 2.8 up until the “Basis for a subspace” section starts.
• You can read Beezer on Subspaces but he does things in a different order and will use fancy words you haven’t seen before. When he says “vector space” pretend he just said $\mathbb{R}^n$.
• Take a look at the solutions for quiz 2.

Slides

#### September 15: Applications of Linear Functions

Slides

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

• Notes § 1.9. and §2.1
• LLM §1.10 and §2.1

#### September 10: Linear Transformations

Slides from Lecture

#### September 8: Solutions and independence of linear systems

Slides from lecture

• Mastery Quiz 1 Due at noon. Submit on Blackboard.
• (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.)
• LLM § 1.5 and 1.7
• Notes § 1.6 and 1.7
• Beezer “Homogeneous Systems of Equations”, “Linear Independence”, and “Linear Dependence and Spans”

#### September 3: Vectors and Matrix Equations:

Slides from lecture

• LLM §1.3-1.4
• 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

Slides from lecture

• Read the introduction in the course notes.
• Watch Essence of Linear Algebra, Chapter 1 by 3Blue1Brown
• Read at least one of the following:
• Sections 1.1-2 of Lay, Lay, McDonald
• Sections “What is Linear Algebra?”, “Solving Systems of Linear Equations”, and “Reduced Row-Echelon Form” in Beezer
• Sections 1.1-3 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.

## Mastery Quizzes

The topics for the quizzes are:

1. Systems of Linear Equations
2. Vector Equations and Spans
3. Linear Independence
4. Linear Transformations
5. Matrix Multiplication
6. Matrix Inverses
7. Subspaces
8. Basis and Dimension
9. Vector Spaces and Subspaces
10. Vector Space Linear Transformations
11. Bases and Coordinates
12. The Matrix of a Linear Transformation
13. Eigenvectors and Determinants
14. Characteristic Polynomials and Finding Eigensystems
15. Complex and Generalized Eigenvectors
16. Change of Basis
17. Similarity and Trace
18. Diagonalization
19. Dot Product and Projection
20. Inner Products
21. Orthogonal Decomposition

## Tests

Graphing calculators will not be allowed on tests. Scientific, non-programmable 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 (ISBN-13: 978-0321982384). 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.