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,...,vn). 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