Strang's Linear Algebra Book vs. The MIT OCW Videos — How to Actually Use Them
Gilbert Strang's Introduction to Linear Algebra is one of the most widely used textbooks in undergraduate math, and the companion video lectures on MIT OpenCourseWare are freely available. That combination is genuinely one of the better free resources you can get, but it's easy to misuse both if you just binge-watch without a plan. Strang teaches from the perspective of column spaces, rank, eigenvalues, and SVD — not from the abstract vector-space definitions you'll find in pure math texts. He builds intuition first, then formalizes. That ordering is what makes the material stick for engineers and data scientists. Most other books lead with definitions and leave the "why" for chapter seven. The book's structure hinges on four fundamental subspaces: column space, null space, row space, and left null space. If you understand how those four interact, everything else in the course clicks. Strang circles back to them constantly. That repetition is deliberate and you should lean into it rather than skim past it.
How I Used This Resource When I Was Learning It
I went through Strang's course about five years ago while working on a numerical methods project. The videos alone weren't enough for me. I needed to do the problem sets. The book's exercises are where the actual learning happens, not the lectures. I'd watch a 50-minute video, then spend 90 minutes to two hours on the problems. Some weeks the ratio was worse. Problem 3.4.12 from the 5th edition took me an afternoon because I kept second-guessing my understanding of pivots in elimination. Here's the thing nobody mentions: the homework problems in Strang's book are harder than they look. They're not computational drills. They're designed to make you think about what the operations are actually doing. If you're doing them in under ten minutes per problem, you're probably missing the point.
What Most People Get Wrong
The biggest mistake I see is treating the video lectures as the primary source of learning. They're supplementary. The book and the problem sets are the work. Watch the videos to get the framework, read the chapters for the details, and do the problems to internalize it. Do not reverse that order. Another trap is skipping the sections on positive definite matrices and SVD because they feel advanced. Those topics show up everywhere in applied work. Eigenvalues and SVD aren't optional chapters. They're the core of the subject. I ran into a specific issue when I was using the matrix factorizations in practice. I was working with a nearly singular matrix — a covariance matrix from some sensor data that had two variables almost perfectly correlated. The theoretical rank was full, but numerically it was effectively rank-deficient. Standard inversion blew up. What I ended up doing was computing the SVD and dropping the components whose singular values fell below a threshold relative to the largest one. That gave me a stable pseudo-inverse solution. It's exactly the kind of edge case the course prepares you for, even if the textbook doesn't walk through this exact scenario.
Get the Full Details
Parsing the Material Efficiently
Start with Chapter 1. Matrix multiplication as combinations of columns. This is the single most important conceptual shift the book makes. If you understand Ax as a linear combination of columns of A, the rest of the course is easier. Don't rush past this. Chapter 2 covers elimination and factorization. Get comfortable with LU decomposition. It's the backbone of nearly every numerical linear algebra application. The derivation is straightforward but the implications are deep. Chapters 3 and 4 are the fundamental subspaces and determinants. Determinants are less important computationally than Strang presents them, but you need the conceptual foundation. Move through them but don't get stuck on determinant-heavy proofs. In practice, nobody computes determinants for anything beyond small matrices.
Chapters 5 through 7 are eigenvalues, differential equations, and Fourier transforms. These connect to applications. If you're studying linear algebra for machine learning or signal processing, these chapters matter more than the determinant chapter. Prioritize accordingly.
Where the Resource Falls Short
Strang's book doesn't cover computational linear algebra from a programming perspective. There's no emphasis on numerical stability, condition numbers, or implementation details. If you want to actually code these algorithms, you'll need supplemental material. Trefethen and Bau's Numerical Linear Algebra fills that gap well, though it's denser. The problem set solutions are available online but not officially from MIT. Some are correct. Some contain errors. Cross-reference your answers when something doesn't check out. I learned to verify my work by implementing small problems in Python and comparing outputs, which is probably better practice anyway. The MIT OCW lecture schedule doesn't always align perfectly with the book's chapter order. The video numbering changed between course iterations. The 18.06 recordings from the early 2000s are the most commonly referenced and they map reasonably well to the 4th and 5th editions of the textbook, but there are minor mismatches. Don't stress over exact alignment between video number and chapter. The content is consistent even if the sequence isn't.
What You Actually Need
The textbook: Gilbert Strang, Introduction to Linear Algebra, 5th edition. It's available through most retailers. The 4th edition is substantially similar and cheaper if you don't need the new problems. The lectures: available free at ocw.mit.edu under course 18.06. No account required. No payment. Just pick the section and start watching. The problem sets: each chapter in the book has them. Do them. All of them if you can. The later chapters have fewer problems but they're still worth doing.
A coding environment: Python with NumPy and SciPy will serve you well. Implementing Gaussian elimination from scratch, even when it's redundant, forces you to understand what's happening under the hood. I did this during the course and it made a real difference in retention.
The Real Takeaway
Strang's approach to linear algebra is practical. It's built for people who need to use matrices, not people who need to prove theorems about them. The course will make you competent, not theoretical. That's its strength and its limitation. If you need pure mathematical rigor, look elsewhere. If you need to understand what a matrix actually does, this is one of the best places to learn it. The subject is cumulative. You cannot skip ahead meaningfully. Each concept builds directly on the previous ones. If you fall behind on elimination and factorization, eigenvalues will make no sense. Keep up with the problem sets. That's the actual work. The videos are the framing device.
