TITLE: Real life applications of general vector spaces
QUESTION [20 upvotes]: Students familiar with Euclidean space find the introduction of general vectors spaces pretty boring and abstract particularly when describing vector spaces such as set of polynomials or set of continuous functions. Is there a tangible way to introduce this? Are there examples which will have a real impact? I would like to introduce this in an engaging manner to introductory students. Are there any real life applications of general vector spaces?

REPLY [2 votes]: Well you could talk about the word vectors? Or even thought vectors, really any time you want a categorical piece of data to be represented in a unique relational database, you can give that piece of data numerical "features", or in other words, dimensions that allow one to classify people, places, words, things, literally anything you want. 
For example Netflix vectorizes movies, and they actually then insert the user as a vector into the same vector space as the movies to get an idea of what other movies to suggest to the user. Vectors are heavily used in machine learning and have so many cool use cases. 
What's really awesome about those kinds of examples is you don't really have to understand the linear algebra to even really understand what is going on, at least from a conceptual point of view. Geoff Hinton (one of the inventors of back-propagation) does work with thought vectors, I really suggest reading up on that because the applications are literally endless. Like Google is developing an algorithm that can flirt with thought vectors. They already have a program that can answer email for you based on the same technology. 
Hope this helps!