Why study finitedimensional vector spaces in the abstract if
they are all isomorphic to R^{n}?
Here are several (closely related) reasons.
 Thinking of a vector space as R^{n} 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 `noncanonical',
and therefore offensive to the delicate aesthetic sensibilities
of the pure mathematician.
 There are many important examples throughout mathematics
of infinitedimensional vector spaces. If one has
understood finitedimensional spaces in a coordinatefree
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 coordinatefree approach. For example, it is a lot easier
to write (and read) v than
(v_{1},v_{2},...,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 infinitedimensional (which simply
means not finitedimensional). 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 finitedimensional vector space V. Start with the
space R^{3} and let V be the subspace consisting of
all vectors (x_{1},x_{2},x_{3}) such
that x_{1}+x_{2}+x_{3}=0. This subspace
is twodimensional, but there is no single basis that stands out
as being the most natural. This example can of course be
generalized to subspaces of R^{n} defined by
simultaneous equations.