What Is Linear Algebra?

Linear algebra is a branch of mathematics that focuses on the study of vectors, matrices, and linear transformations, forming the foundation for understanding multidimensional spaces and solving systems of equations. Its applications are vast and transformative, spanning fields like computer science, engineering, physics, economics, machine learning, and more. Linear algebra powers technologies like 3D graphics, artificial intelligence, and optimization algorithms, and it is crucial for modeling real-world phenomena such as fluid dynamics, financial markets, and quantum mechanics. By learning linear algebra, individuals gain the tools to think abstractly, solve complex problems, and unlock innovations in science and technology, making it an essential skill for anyone looking to thrive in a data-driven, interconnected world.

Who Is The Course For?

The course is mainly intended for undergraduate science and engineering students. It can also be used as a refresher by science and engineering professionals. Last but not least, anybody who wants to really understand Data Science and Machine Learning must master Linear Algebra first. In any case, keep in mind that this is not some lightweight video-based MOOC. It’s rigorous one-semester college-level course that is not easy, and that requires a significant amount of effort and time to complete.

How Does The Self-Paced Course Differ from Traditional Lectures?

Students work at their own pace in a gamified environment with the help of bite-sized tutorials, examples, exercises, and graded practical tasks and quizzes. The instructor does not lecture, and is helping each student individually instead. Throughout the entire course, students receive instant AI-based feedback from the software platform. Thanks to its self-paced nature, the course contains more review material, and has a more gentle learning curve than traditional lectures. The course provides students with much more hands-on practice than the traditional lecture + homework model, and its format is excellent for remote instruction. Students are exposed to entry-level scientific computing with Linear Algebra via Python and Numpy. Year after year, the course is constantly being enormously popular with students. More details about the course and the teaching method can be found in the peer-reviewed paper below.

Recommended Background

To succeed in this course, you should know basic high school math including arithmetic, working with fractions, relations, and functions. You should also know trigonometric, exponential, and logarithmic functions. Towards the end, the course uses inner products of polynomials which requires very basic knowledge of integration.

Student Learning Outcomes (SLO)

Students will be able to:

  • Perform all standard vector and matrix operations.
  • Transform linear system to matrix form and solve them.
  • Create echelon and reduced echelon forms of matrices.
  • Determine basic and free variables in linear systems.
  • Express infinite solution sets in parametric vector form.
  • Work with determinants and elementary matrices.
  • Create and use LU and Cholesky matrix factorizations.
  • Work with linear transformations associated with matrices.
  • Calculate eigenvalues and eigenvectors.
  • Work with linear spaces, bases, and coordinates.
  • Perform orthogonal projections and decompositions.
  • Diagonalize matrices.
  • Perform spectral decomposition of matrices.
  • Solve Least-Squares problems via normal equations and QR factorization.
  • Perform Singular Value Decomposition (SVD) of matrices.
  • Apply SVD to data cleaning and image compression.

Some of the more advanced material towards the end of the course can be made optional or left out.

Equipment Requirements

Computer, laptop or tablet with Internet access, email, and one of the following browsers:

  • Google Chrome
  • Mozilla Firefox
  • Microsoft Edge
  • Safari

Matrix App

The course uses an innovative proprietary Matrix App which allows students to quickly manipulate matrices and augmented matrices on desktops or touchscreen devices with their mouse or fingers, respectively. The following short video illustrates how the Matrix App is used to enter an augmented matrix and obtain its reduced echelon form:

The Matrix App is described in great detail in the peer-reviewed paper below.

Syllabus and Course Details

Click here to access the syllabus. Use the links in the table of contents to see how the various topics that you will be expected to master.

Time Commitment

Keep in mind that this is a full-semester college-level course whose completion requires a significant commitment. You will be working actively 100% of the time, not just watching some video tutorials.

Pricing

A single-user license costs $2400 and most of the fee goes towards 12-month individual remote tutoring which is crucial for your success. School licenses are available at a substantially reduced rate which depends on the number of students taking the course.

Hear From a Student

References

P. Solin: Self-Paced, Instructor-Assisted Approach to Teaching Linear Algebra. Mathematics in Computer Science 15(4):1-27 (2022). Download full paper in PDF.

This peer-reviewed paper in Mathematics in Computer Science describes in great detail the course itself, as well as the self-paced, instructor-assisted teaching method. It provides guidance to instructors who use the self-paced course material, and presents the results of a quantitative analysis of students’ performance, results of a student survey, and student testimonials.

Interested in Learning More?

If you are interested in using this course material with your students, or in learning Linear Algebra yourself, then please fill out the form below, and we will reach out to you to get you on your way.