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.