Introductory Linear Algebra
SAT Score Range
•
14 sessions
•
DG
AS
SV
+27
This series ended on August 2, 2022. All 1:1 and group chats related to this series are disabled 7 days after the last session.
About
Linear algebra began as the study of linear systems of equations. Currently, linear systems remain of great importance, in part due to their relative simplicity. For true mastery in any field, from physics to computer science, an understanding of linear algebra is fundamental. One might even argue that linear algebra and its applications are even more important than those of calculus in today's world.
This course provides an introduction to linear algebra from a concrete point of view. We emphasize understanding through intuitive exposition and plenty of examples and applications. Whilst we will need to understand theory, we will focus on the concrete setting of Euclidean space and relegate more abstract ideas such as general vector spaces to a second course.
Co-taught with SHW tutor @Yu Heng L, who also developed the series idea, overview, and outline!
PREREQUISITES
High school Algebra I and II are required. Pre-calculus, trigonometry, and calculus are not necessary.
SYLLABUS
1. Linear Systems (~5 sessions)
2. Matrices (~3 sessions)
3. Determinants (~2 sessions)
4. Eigenvalues and Eigenvectors (~4 sessions)
Tutored by
Schedule
✋ ATTENDANCE POLICY
Please do your best to show up to all sessions, as all covered concepts build on each other. If possible, let the instructor(s) know ahead of time if you must miss a day so they can help you catch up in a timely manner.
SESSION 1
21
Jun
SESSION 1
Orientation
Orientation
Tue 1:00 AM - 2:30 AM UTCJun 21, 1:00 AM - 2:30 AM UTC
DAY 1
We'll introduce ourselves and our interests in math, as well as what we expect and hope to take away from this series.
We'll introduce the concept of systems of strictly linear equations and the fundamentals of matrix notation. We'll also delve into using row reduction and entry elimination (Gauss-Jordan Elimination) to simplify and solve systems in matrix notation.
SESSION 2
23
Jun
SESSION 2
Other Topics
Other Topics
Thu 1:00 AM - 2:30 AM UTCJun 23, 1:00 AM - 2:30 AM UTC
DAY 2
We'll introduce using row reduction to simplify matrices and two possible and common forms a matrix can take on: echelon form and reduced echelon form.
SESSION 3
25
Jun
SESSION 3
Other Topics
Other Topics
Sat 1:00 AM - 2:30 AM UTCJun 25, 1:00 AM - 2:30 AM UTC
DAY 3
We'll go over common types of vectors, basic matrix operations, types of variables, and consistency. We'll also go over fundamental algebraic properties of matrices. We'll then introduce the most powerful and versatile equation we'll be using in the rest of the series: the matrix equation Ax = b.
SESSION 4
28
Jun
SESSION 4
Other Topics
Other Topics
Tue 1:00 AM - 2:30 AM UTCJun 28, 1:00 AM - 2:30 AM UTC
DAY 4
We'll talk through the concept of linear transformations and the matrices and graphs of common transformations.
SESSION 5
1
Jul
SESSION 5
Other Topics
Other Topics
Fri 1:00 AM - 2:30 AM UTCJul 1, 1:00 AM - 2:30 AM UTC
DAY 5
We'll introduce homogeneous equations and their trivial and non-trivial solutions, writing solutions in parametric form, and linearly independent and dependent vectors.
SESSION 6
5
Jul
SESSION 6
Other Topics
Other Topics
Tue 1:00 AM - 2:30 AM UTCJul 5, 1:00 AM - 2:30 AM UTC
DAY 6
We'll finish up Unit 1 (Linear Systems) by exploring common real-world applications of linear systems and matrices, such as in balancing chemical equations, analyzing traffic flow, and observing population change over time.
SESSION 7
8
Jul
SESSION 7
Other Topics
Other Topics
Fri 1:00 AM - 2:30 AM UTCJul 8, 1:00 AM - 2:30 AM UTC
DAY 7
We'll delve into newer matrix operations and their practical applications.
SESSION 8
12
Jul
SESSION 8
Other Topics
Other Topics
Tue 1:00 AM - 2:30 AM UTCJul 12, 1:00 AM - 2:30 AM UTC
DAY 8
We're halfway through! We'll introduce methods of calculating the inverse of a matrix. We'll also go over the Inverse Matrix Theorem (IVT), which will be crucial to the remainder of the course.
SESSION 9
15
Jul
SESSION 9
Other Topics
Other Topics
Fri 1:00 AM - 2:00 AM UTCJul 15, 1:00 AM - 2:00 AM UTC
DAY 9
We'll walk through factorizing a matrix by its upper and lower triangular forms and how this can help us with more advanced matrix operations in the future.
SESSION 10
19
Jul
SESSION 10
Other Topics
Other Topics
Tue 1:00 AM - 2:30 AM UTCJul 19, 1:00 AM - 2:30 AM UTC
DAY 10
We'll begin unit three by introducing the concept of a matrix determinant, its importance, and different methods of calculating it. We'll also go over some fundamental properties of a determinant.
SESSION 11
22
Jul
SESSION 11
Other Topics
Other Topics
Fri 1:00 AM - 2:30 AM UTCJul 22, 1:00 AM - 2:30 AM UTC
DAY 11
We'll go over different methods of using determinants to solve a system, including Cramer's Rule and the adjoint/adjugate. We'll also use matrices and determinants to calculate the area of a two-dimensional parallelogram and the volume of a three-dimensional parallelepiped.
SESSION 12
26
Jul
SESSION 12
Other Topics
Other Topics
Tue 1:00 AM - 2:30 AM UTCJul 26, 1:00 AM - 2:30 AM UTC
DAY 12
We'll spend this session perusing a more abstract field of linear algebra to understand basic novel concepts and the important roles they will play in the following sessions. We'll go over bases and different types of matrix spaces, and what they can reveal about a matrix.
SESSION 13
29
Jul
SESSION 13
Other Topics
Other Topics
Fri 1:00 AM - 2:30 AM UTCJul 29, 1:00 AM - 2:30 AM UTC
DAY 13
We'll begin our fourth and final unit on eigenvectors and eigenvalues by introducing their concepts and importance, as well as using the Characteristic Equation to relate the two.
SESSION 14
2
Aug
SESSION 14
Other Topics
Other Topics
Tue 1:00 AM - 2:30 AM UTCAug 2, 1:00 AM - 2:30 AM UTC
DAY 14
We'll go over diagonalizing matrices to be able to easily compute high powers of matrices. We'll also introduce Markov chains and stochastic modeling, and then wind down the session with group discussion and series feedback.