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?
Many years ago I was having a beer with a couple of fellow math grad students at some place around Harvard Square, and we overheard some guy at the next table trying to impress a girl telling her that he was taking a Linear Algebra course which was "so difficult" having to deal with spaces of "many dimensions".
I am not sure that this Linear Algebra technique of picking up girls at a bar may be listed among "real life applications", but, if it works, sure offers an important motivation.
What's boring about polynomials and real-valued functions ?
Polynomials have a great use in science, mainly in approximations using interpolations.
Since the set of polynomials with degree smaller than $n$ is a vector space, we can take an orthonormal basis for it and easily find approximation for any real value function (depending on the inner product of course). note that the reason we can do this is that the real valued functions are also a vector space!
Depending on how much depth you want to introduce, I think you should mention fourier analysis. Even if they haven't taken differential equations courses before, showing that functions form a vector space is quite trivial. Once you have this, you know you can introduce the idea of a basis for this space, which lets you reliably decompose certain functions as being made up from other ones. Applications obviously abound. Probably not "obviously" for your students, but take your pick to wow them, since it probably has a use in whatever they're studying.
The point is that this concept of "representing a function as a sum of others" would be difficult to define as a concrete method with only the tools they've seen so far. But it's the same ideas as vector projection and finding a basis, but on things that don't look like vectors at all. The abstract idea of vector spaces lets you carry all these neat and powerful tools over to other problems.
Is the stock exchange real enough? OK, you'll have to abstract from the fact that you only can buy or sell complete stocks, not arbitrary fractions of stocks (although given the modern way of trading stocks, I'm not even completely sure if you really can't ;-)). But it is clear that if you e.g. put an order to sell two Microsoft stocks and buy one Apple stock, and then put another order to buy two Apple stock and five IBM stock, you ultimately sell two Microsoft stock and buy three Apple and five IBM stock. So you've got vector addition (-2,1,0,0,....) + (0,2,5,0...) = (-2,3,5,0...). This also has the advantage that there's no natural order of your basis (there's no reason why you should list Microsoft stock first, or even why you should put all stock in a linear order). Multiplication by a scalar is also clear: You buy/sell e.g. twice as many stocks.
You also get a natural dual space: The price. It is a mapping from a vector (sell/buy order) to a scalar (money to pay/earn), and it is obviously linear (if you buy one Apple and two Microsoft stocks, you pay the price of an Apple stock plus twice the price for a Microsoft stock. Stock exchange covectors are regularly listed in certain newspapers and on certain web sites.
Also it has the advantage of not having an intrinsic scalar product (you can't meaningfully ask about the product of the order "selling two Microsoft stocks" and the order "buying one Microsoft stock and selling three Apple stocks"). When coming from Euclidean vector spaces, you are inclined to take a scalar product for granted, so having a space where a scalar product simply doesn't make sense is probably a good idea.
I don't know if this is what you are looking for, but...
The functioning of the 4G-smartphones depends on the phones ability to quickly carry out certain transformations (DFT/IDFT) in certain (for example) 1024-dimensional subspaces of the space of (periodic) functions.
Look up OFDM for more details.
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!