Why study finite-dimensional vector spaces in the abstract if they are all isomorphic to Rn?

Here are several (closely related) reasons.

Let me give two examples of vector spaces that illustrate some of the above points. First, the set of all continuous functions defined on the closed interval [0,1] can be made into a vector space in a very natural way. This vector space is infinite-dimensional (which simply means not finite-dimensional). If you do not immediately know how to prove this then it is a good exercise.

Of more relevance to the question that heads this page is the following finite-dimensional vector space V. Start with the space R3 and let V be the subspace consisting of all vectors (x1,x2,x3) such that x1+x2+x3=0. This subspace is two-dimensional, but there is no single basis that stands out as being the most natural. This example can of course be generalized to subspaces of Rn defined by simultaneous equations.