Working Through Linear Algebra With Strang
I ran into a wall trying to make sense of matrix factorizations during a project last year. My code was choking on a 500x500 sparse system that should have been trivial, and I realized I understood the mechanics but not the why behind them. Went back to Strang's MIT OCW lectures. Finished the course about six months later. Here is what I learned along the way. The course lives on MIT's OpenCourseWare site, and you can find it by searching for "linear algebra gilbert strang" or going straight to ocw.mit.edu. The lectures are organized into 35 sessions, each roughly an hour long. They are available for free as video recordings with accompanying PDF notes and problem sets. The textbook "Introduction to Linear Algebra" by Gilbert Strang accompanies the lectures but is not required to follow along. The big thing that separates this from most linear algebra courses is that Strang spends actual time on the geometric intuition behind the operations. Most textbooks introduce Gaussian elimination, then immediately pivot to abstract vector spaces. Strang does the reverse. He makes you see what matrix multiplication actually does to space before asking you to prove anything about it. That shift in perspective matters more than people realize.
Here is the structure: the first dozen lectures cover the fundamentals—systems of equations, matrix elimination, column spaces, determinants. Then he moves into applications, eigenvalues, singular value decomposition, and Fourier transforms. The SVD section alone is worth the entire course. It is the single most useful tool you will encounter in applied mathematics, and Strang explains it in a way that does not require a math degree to follow. One issue people run into is that the problem sets are genuinely difficult. I went through Lecture 7 on determinants and got stuck on Problem 3 for about two hours. The answer in the back of the book is correct, but the path to it is not obvious from the lecture material alone. I had to rewatch that segment three times and sketch out the geometric interpretation on paper before it clicked. This happens more than once. It is not a flaw in the course. It is the course doing its job. There is a common misconception that you need to be strong in calculus to take this course. You do not. The linear algebra stands on its own. Calculus appears later when Strang connects things to differential equations, but the core material assumes nothing beyond high school algebra. I met a grad student in my lab who had never taken a formal linear algebra class and completed this course in a summer. She is now a data scientist.
Another thing that catches people off guard is the pace. Each lecture is dense. Strang moves quickly between topics and expects you to fill in gaps yourself. When he introduces the four fundamental subspaces in Lecture 18, he assumes you already internalized everything from Lectures 2 through 16. If you skimp on the earlier material, Lecture 18 becomes nearly impenetrable. I learned this the hard way after pushing through the first half too quickly. Had to loop back and redo three weeks of content before I could keep up. The supplementary materials are worth using. The chapter summaries at the end of each section in the textbook condense roughly an hour of lecture into two pages. I used those as review sheets before testing myself on the problem sets. The YouTube channel also has Strang's full classroom recordings, which sometimes help when the OCW audio quality is thin. If you are taking this for engineering or physics, pay close attention to the sections on orthogonal matrices and projections. Those show up everywhere. If you are going into machine learning, the SVD and least squares lectures are non-negotiable. I wasted about forty minutes last month debugging a numerical instability that traced back to a badly conditioned matrix—the exact scenario Strang warns about in Lecture 24. Having seen that warning ahead of time would have saved me the trouble.
Get the Full Details
One practical tip: do not watch the lectures passively. Work through at least one problem set per lecture before moving forward. The difference between someone who understands linear algebra and someone who memorizes it comes down to doing the problems, not watching the videos. I made that mistake early on and fell behind fast. The course is free. The MIT OCW page has direct links to all 35 lectures, the downloadable notes, and the problem set solutions. There is no certificate unless you go through MITX online, but that costs money and most people do not need it. The knowledge is what matters.
What Makes This Approach Different
Strang treats matrices as operators on vector spaces rather than collections of numbers. That is a small wording difference but it changes how you approach every problem. When you stop thinking of a matrix as just an array and start thinking of it as something that transforms space, the abstract concepts become concrete. Eigenvalues stop being mysterious and start being about stretch factors. Determinants become about volume distortion. It is a different way of seeing the same math, and it tends to stick. The four fundamental subspaces framework—that is Strang's signature contribution to how this subject is taught. Column space, null space, row space, left null space. Learning to identify all four for any given matrix and understand how they relate to each other gives you a mental model that applies to differential equations, optimization, signal processing, and machine learning. Most courses touch on this. Strang builds the entire second half of the course around it. There are some limitations to be aware of. The course does not cover numerical linear algebra in depth. If you want to understand how to actually implement matrix operations in code without running into floating point issues, you will need to supplement this with something like Trefethen and Bau. The theoretical foundations here are solid, but the computational side is lighter than what a computer science student might need. I picked up the rest from a separate course on numerical methods and ended up with a much more complete picture.
Another gap is the treatment of complex vector spaces. Strang mentions them briefly but does not develop the theory. If you are working with quantum mechanics or advanced signal processing, you will need to fill that in elsewhere. For everyone else, the real-valued framework covers the vast majority of applications. The problem sets are also known for being harder than what you find in standard textbooks. They are designed to make you think, not to give you procedural practice. Some problems require insights that are not explicitly taught in the lectures. This is good for deep understanding and frustrating in the moment. I recommend spending no more than an hour on a single problem before checking hints or discussing with others. Going longer than that usually means you are overthinking it rather than working through it properly. If you are deciding whether to invest time in this course, the answer is yes unless you are already comfortable with all of this material. The lectures are available for free, so the cost of trying is zero. The return on investment is high for anyone working in a technical field. I wish I had encountered this course sooner, but it is better late than never.
