Also, if you transform bag-of-words vectors into a dense form, you're gonna have a bad time (insert appropriate meme picture here). In large corpora, dimensionality grows quite substantially - if you work with news corpora or Wikipedia, you're in the 100k-1M dimensional space pretty quickly.
Great to see an approachable explanation for NLP. As they say sometimes, when you know how it's done, it stops being "Artificial Intelligence".