Generalization of Lines, Planes (Finite Fields)

[Bacle]

Hi, all:

Say we have a bare-bones Vector Space v, i.e., V has only the basic vector space

layout; no inner-products, etc., over a finite field .

I think then , we can still define a line in V as the set {fvo: vo in v, f in F}, i.e.,

as the set of all F-multiples of a fixed vector vo in V .

Is there a way of generalizing the notion of a plane to these vector spaces?

Thanks in Advance.

 

[micromass]

Yes, of course. This is possible in any vector space over ANY field $$\mathbb{K}$$. A plane (through the origin) is simply defined as the span of two linear independent vectors. This definition makes sense for any field $$\mathbb{K}$$, be it the rational, reals, complexes or finite fields...