> 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 d

> 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.