Hi everyone,
In this series we'll go over a full college-level linear algebra course covering matrix operations, linear independence, vector spaces, various decompositions (LU, CR, QR, diagonalization, SVD), determinants, eigenvalues and eigenvalues, as well as markov chains and systems of differential equations, among other topics.
We'll meet Mondays for about 90 minutes starting at 7:00 PM ET. Hopefully, I do not have to use the full time every session, but plan on utilizing the full time. Note that we will NOT meet on Monday, January 20 (MLK Day) nor Monday February 17th (Presidents Day), as well as for two weeks in March during my Spring Break. As much as I want to share Math, I also want to enjoy the end of my senior year in High School. I have all the sessions for the months of January and February loaded in right now, and I'll add more as needed.
This is NOT intended to replace an equivalent 200-level Applied Linear Algebra college course, but rather is intended either as a supplement or for any curious high-schoolers to learn some math for fun and get some exposure to Linear Algebra. I want to prioritize a broad overview of the topics rather than getting into the weeds of mathematical rigor (breadth over depth).
Completion of Calculus is not required but is strongly encouraged. Learners should be concurrently enrolled in some form of Calculus, however. Some basic familiarity with matrices, vectors, and matrix/vector multiplication is also preferred. Learners are expected to be proficient in Algebra 2 and Pre-Calculus.
Lectures from the Spring 2024 Linear Algebra course which I took at a nearby four-year university.
Homework problems are included at the bottom of every section for extra practice, though they lean more on the theoretical side. Ignore all the MATLAB projects as we will not be doing them (I think Image Clipping is the first one). I may provide practice exams in case you guys wanted to see what a typical exam could look like for this class.
I want to offer take-home exams and quizzes. Because this does not count towards credit towards any institution, cheating on them would only be doing a disservice to yourself and I trust that everyone completes these honestly. We should have a quiz for most sections and have three practice exams. While I don't have many large blocks of time to host sessions, I'm more than open to take questions via DM throughout the day.
We will cover the following sections in this tentative order, assuming sufficient interest is sustained:
Unit 1: Introduction and the Fundamental Subspaces:
1.1 Introduction & Linear Combinations
1.2 Linear Systems
1.3 & 1.4 Gaussian Elimination
1.5 Parametric Vector Form
1.6 Matrix Multiplication
1.7 Matrix Inverse
2.5 Linear Independence
2.6 Four Fundamental Subspaces
1.8 LU & CR Factorization
Take-Home Test
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Unit 2: Determinants and Eigenvalues:
1.9 Linear Transformations
2.1 Families of Matrices
2.2 Determinants
2.4 Vector Spaces & Orthogonality 2.3 & 2.7 Eigenvalues and Eigenvectors
2.10 Diagonalization
2.8 Orthogonal Projection & Least Squares Solutions
2.9 QR Factorization & Orthonormalization
Take-Home Test
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Unit 3: Applications of Linear Algebra:
3.3 Orthogonal Diagonalization
3.4 & 3.5 Singular Value Decomposition
3.1 Discrete Linear Dynamics (Markov Chains)
3.2 Continuous Linear Dynamics (Systems of 1st-order Differential Equations)
Take-Home Final Exam
