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

Here are several (closely related) reasons.

• Thinking of a vector space as Rⁿ encourages us to think of an individual vector as a string of numbers. It is often more illuminating, however, to think of a vector geometrically - as something like a magnitude and a direction. This is true particularly with vectors that come from physics.
  
• To turn such a vector into a string of numbers one must first choose a coordinate system, and very often there is no choice that is obviously best. In such circumstances, choosing coordinates is necessarily `unnatural' and `non-canonical', and therefore offensive to the delicate aesthetic sensibilities of the pure mathematician.
  
• There are many important examples throughout mathematics of infinite-dimensional vector spaces. If one has understood finite-dimensional spaces in a coordinate-free way, then the relevant part of the theory carries over easily. If one has not, then it doesn't.
  
• There is often a considerable notational advantage in the coordinate-free approach. For example, it is a lot easier to write (and read) v than (v₁,v₂,...,v_n). To give another example, a simple looking equation like Av=b can turn out to stand for a system of m equations in n unknowns.
  

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.