*En passant*, in a standard text such as Herstein's *Topics in Algebra*, you'll find a definition of vector space (corresponding to the one in your notes) but also, and more importantly, a host of non-trivial examples that go beyond n-tuples. For example, spaces of functions. A matrix can be considered as a linear operator on elements of a vector space. The set of such linear operators (e.g., matrices) is itself a vector space (in fact, a bit more, as it's both a vector space and a ring, making it an "algebra"). On the space of differentiable functions, the derivative can be considered a linear operator. In short, the terminology of vector spaces subsumes n-tuples and matrices.