What Professor Leonard Linear Algebra Actually Covers
If you're trying to learn linear algebra and you end up on Professor Leonard's channel, you've probably got a long playlist ahead of you. His course runs roughly 30+ hours across vectors, matrices, determinants, eigenvalues, and the later applications stuff. It's complete in the sense that it takes you from no background to a full semester of content. The pacing is slow on purpose. He writes out every step on the board. Start by watching the first lecture on vectors. Don't binge more than two videos in a sitting. His explanations are thorough but he repeats himself, and once the novelty wears off you'll zone out around the 45-minute mark and miss the actual mechanism he's walking through. Take notes in your own words. Not transcription, just enough that you can reconstruct the argument later without rewinding. The real value is in the example work. He does problem after problem in sequence. Skip the ones that feel obvious to you, but don't skip them all. The problems build on each other.
When you hit row reduction and augmented matrices, slow down. That's where most people fall apart, and Leonard is at his clearest here. Gaussian elimination, reduced row echelon form, free variables versus pivot variables. Watch him do it twice. Once when it's simple, once when the numbers get annoying. You need both.
Where the course actually shines
His strength is in building definitions from concrete examples rather than dropping notation and pretending you already know what it means. When he gets to span, linear independence, and basis, he starts with geometric intuition in R2 and R3 before moving to abstract vector spaces. That transition is usually brutal in textbooks. Leonard makes it bearable. The eigenvalue section is similarly patient. He derives why we care about Av = v instead of just stating the characteristic equation and moving on. The geometric interpretation of eigenvectors as directions that don't rotate during a transformation is worth three full lectures because most courses gloss over it in twenty minutes.
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Where it falls apart
Here's the thing nobody on the internet will tell you: his course assumes you've already taken College Algebra or Precalculus and are comfortable manipulating expressions, factoring polynomials, and working with exponents without pausing to think about it. If you struggle with algebraic manipulation, linear algebra will eat you alive regardless of which instructor you follow. Leonard does not remediate algebra. He moves past it quickly and expects you to keep up. The video length is also a structural problem. At roughly 45 to 60 minutes per lecture, this isn't course material you can watch during a lunch break. You need dedicated blocks of time. A lot of dedicated blocks. If you're studying for an exam in two weeks, this isn't your resource. It's a semester-long commitment, not a cram tool. There's also no accompanying textbook, no problem sets with answers, and no interactive component. You watch him solve problems. You don't actually solve them yourself unless you pause and do it on paper. That self-testing step is where learning happens, and it's entirely on you to enforce it. Most people don't.
A specific issue I ran into
When I was working through the section on diagonalization, I kept getting stuck on exercises where the matrix had a repeated eigenvalue but only one independent eigenvector. Leonard covers the basic case and then moves on to applications like differential equations. He mentions defective matrices in passing but doesn't drill into Jordan canonical form or the general procedure for handling geometric multiplicity less than algebraic multiplicity. If you run into that gap, you'll need to supplement with another source. Strang's linear algebra lectures handle the defect case more directly, though his teaching style is a completely different animals. My workaround was to write out the definition of geometric multiplicity separately, work through three examples where the eigenspace dimension was strictly smaller than the eigenvalue multiplicity, and only then return to Leonard's material. It took about forty-five minutes of extra work that the course doesn't provide.
Practical setup recommendations
Use a second monitor or split screen so you can pause and work on paper without losing your place. Notepad or a notebook is better than typing. The act of writing out row operations by hand reinforces the mechanical skill faster than anything else. Also, use a pen that bleeds through slightly less expensive paper. You'll be doing a lot of writing. If your goal is passing a specific university course, check the syllabus first. Some programs emphasize computational methods and will cover more row reduction practice than Leonard does. Others lean theoretical and will want proofs that Leonard skips. His course aligns best with a standard undergraduate computational linear algebra sequence like what you'd find in a typical STEM major requiring the subject.
Download access
The lectures are hosted on YouTube under the Professor Leonard channel. There's no official download from him directly, but the YouTube interface allows offline viewing through the mobile app if you have a Premium subscription. Some universities mirror his content on their own platforms for enrolled students. If you find a direct video link elsewhere, verify it's from the original channel to make sure you're watching the unedited versions with the full explanations intact. If you already know linear algebra and just need a refresher, this will take too long. You're better off with a targeted review document or Strang's short-form lectures. If you're looking for a proof-based treatment, you'll outgrow this quickly. Leonard teaches computation and intuition, not theorem-proof structure. If your course requires proofs, treat this as a supplement, not your primary material. The course also won't help much if your end goal is machine learning applications specifically. The content is there, but the bridge from eigenvalues to PCA or SVD isn't built explicitly in his lectures. You'll need to make those connections yourself or pair the course with a separate applied resource.
For most people taking linear algebra for the first time in a standard undergraduate program, this remains one of the more complete free resources available. The tradeoff is time and the expectation that you will do the practice work yourself. The videos alone won't make you proficient. They give you the framework. Everything else comes from working problems after each lecture.
