Rewatching Strang’s lectures on Linear Algebra after many years amazes me. Right now I am watching his lecture on Matrix multiplication. In just the first 20 minutes, he blew my mind. His explanation on how matrix multiplication works (the five ways) make theorems that may appear look very intuitive.

Columns at a time reminds me of embeddings in low dimensional spaces and specifically Johnson–Lindenstrauss lemma and the improvement by Achlioptas. One key insight of the idea of the embedding is that each column of B in the multiplication A * B gives a testimony (linear combination) of the elements in a row of A. The theorems in these seminal works examine the conditions that should apply to B in order to minimize the distortion of A’s projection to a lower dimensional space. Nevertheless, one key insight for these theorems is explained in the third lecture of a first year course in such a wonderful way!

Columns of A x rows of B make low-rank approximations look obvious how they work. I am pretty sure the other three (classic, row of A at a time, by blocks) ways also contribute to things that didn’t pop to my mind directly.

I would definitely suggest to anyone to (re)watch these lectures.

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I have to admit that your 2nd paragraph goes way beyond my understanding of Linear Algebra, but I found that Strang’s lectures and the lessons in Paul’s Online Math Notes (he has taken down his Linear Algebra section with no explanation, but I’ve reuploaded the full PDF to http://www.mediafire.com/view/?u1u3k2c50qbm84d) were invaluable when I was studying it. They both worked together in making Linear Algebra one of my favorite subjects, even though I decided to study it quite late during my years at the University of Patras.

I know I’ll definitely come across LinAlg in the future, and I’m really looking forward to taking the time to explore its wonders. If you have any recommendations on books or lectures that go just beyond what we were taught at an Undergraduate level (besides some of Strang’s lectures towards the end which do go beyond that), I’d be really interested to look into them.

I am glad you enjoyed these lectures.

I think studying machine learning is a great motivator to dive deeper into linear algebra. Of course, machine learning is not just linear algebra (also statistics, optimization, probabilities) but a lot of interesting problems require background in linear algebra.

So far, the little things that I have studied about linear algebra beyond the courses at the university and Strang are because of Machine Learning. If I find anything interesting, I will update you.

I am also re-watching Strang’s linear algebra videos (currently at the LU factorization). Watched half of the lectures about 2.5 years ago, then school started and actually took a linear algbera class that semester (we used Axler’s “linear algebra done right”). Two years on, re-watching Strang’s videos and seeing the abstract ideas of Axler’s textbook being played out concretely in terms of computation and matrices – it’s amazing! I can now understand the theoretical backing behind what Strang is doing, while gain intuition (and actual computation practicality) on what I learnt from Axler!