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Hey!

Author here. Really appreciate the kind words. If you have any feedback on what I can do to improve the explanation further, I’d love to hear it.

Also, if you’re interested, I’ve written some other posts on explaining concepts in Math and ML following a similar approach:

1. Brief History of CNN based Image Segmentation: https://blog.athelas.com/a-brief-history-of-cnns-in-image-se...

2. Understanding Baidu’s Deep Voice for voice synthesis: https://blog.athelas.com/paper-1-baidus-deep-voice-675a32370...

3. An intuitive explanation of matrices as linear maps: https://dhruvp.netlify.com/2018/12/31/matrices/

Your exposition is very good. If I could make one pedagogical recommendation, it would be that you give a gentle introduction to linear dependence and independence. You don't need this to treat the subject in the geometric fashion you're using (it's implicit in talking about calculating lines on the Euclidean plane). But if you explain it well it's a very powerful didactic mechanic for your audience, because then they can generalize the reason why {(1, 0), (-1, 0)} is not a basis of R^2 to arbitrary dimensions n. It also leads you into the neat result that any linearly independent subset S of a vector space V whose cardinality matches the dimension of V is a basis.

That being said, I know all too well what happens to good exposition when you try to shove too much into it. So feel free to ignore this; you write well and I think you accomplish the goal of presenting the material in an intuitive way.

> The author gives a visual example, for example, of why basic vectors 1,0 -1,0 are bad. The article shows they cannot span the whole space.

I'm assuming you didn't mean the word "basic" this way, but on the off chance you did: {(1,0), (-1,0)} doesn't just not span R^2; it's not a basis of R^2. Partly because it doesn't span R^2 like you've said. But it's also linearly dependent, which is really the more important thing (since every linearly independent subset of a vector space whose cardinality matches the dimension of the space is a basis).

Just wanted to make the terminology of "basis" clear, since you used the term "basic" which could somewhat collide with it.

> On a separate note, I am wondering if such good step by step + counter examples, knowledge presentation -- is a result of author studying at MIT, or a natural trait (or both)?

MIT has very high quality instruction, and the author could be naturally talented at exposition. But I doubt it's because of either of these things. Realistically you could attend any math department in the top 100 and be equally capable of succinctly writing about this topic at the same detail for a general audience the way the author has. Schools like MIT shine on the upper level material and research capital.

Likewise it's probably less about natural talent and more about practice and good editing. On the other hand, what would be very helpful is getting exposure to many textbooks with different pedagogical styles. Someone who learns linear algebra in the hardcore abstract style might have difficulty presenting the material in an accessible, geometric way even if they fully understand it. Similarly if you've only learned about vectors in the Euclidean sense of direction and magnitude, it can be difficult to teach in the more abstract, non-geometric style.

I think what happens is when you spend enough time thinking about explaining it to others or yourself (for whatever reason), you tend to be able to simplify explanations and find creative ways to explain it. Obviously, it is a mixture of raw talent and something else but there is enough raw talent in this world and not enough something else.

You can absolutely visualize why [1, 0] and [-1, 0] is not a basis for R^2! The problem is if you plot them in R^2, then they and the origin all lie on a single line. No matter how you sum copies of these vectors, you can never "escape" this line. Thus, you cannot construct vectors in R^2 that do not lie on this line.

Yes, that's exactly how the author of the article is explaining it (including the picture).

I was just commenting that it was good that, the author took time to explain this.

And those kinds of nuances/counter examples are not always available in explanatory dynamic visualization tools (such as the ones referenced in the comments here [1] ).

To be clear, I am not arguing against dynamic visualizations.

These are very nice, very very helpful, and very time consuming to build and present on a website.

I am just suggesting that showing negative examples is a complimentary, and powerful explanation technique.

[1] http://setosa.io/ev/eigenvectors-and-eigenvalues/