It is better because it really covers every step of the construction process.
And offers explanation of why certain thing are not the right construction blocks. 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.
Those kinds of explanations of 'bad constructions' are difficult to show in visualizations, that show 'good' constructions only.
But, yet, in my view, these negative examples, are really helpful to explain the material that otherwise, requires 'intuition'.
Not everybody has same intuition, so showing negative examples/impossible constructions, and why those do not work -- is a good way tuning one's intuition.
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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) ?
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/
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.
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 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.
It gets weird when thinking of 2D rotations though... Too complex for me!
Personally, my "intuition" is based on analogies with non-complex eigenvectors, and experience solving eigenvector problems algebraically without using pictures.
Also, your 3d rotation example actually has three eigenvectors, two of which are complex. You've only found the one that's real.
Your linear map A moves things around, and you aim to characterise the linear map.
So, look for lines (through the origin) that are not moved. Those are given by eigenvectors. A point on that line might be moved closer to or further away from the origin (depends on eigenvalue < or > 1), or even flipped to the other side (if eigenvalue < 0), but the line as a whole is mapped to itself.
You can drag things around and change values -- if you're a visual learner, it really helps grasp things like this.
https://www.wolframalpha.com/input/?i=stream+plot+%7B-5x+%2B...
The eigendirections are the directions where the solution moves in a straight line.
Not all matrices have (real valued) eigenvectors:
https://www.wolframalpha.com/input/?i=stream+plot+%7Bx+%2B+5...
