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 coordin
