Examples of Vector Spaces - Finite Vector Spaces

Finite Vector Spaces

Apart from the trivial case of a zero-dimensional space over any field, a vector space over a field F has a finite number of elements if and only if F is a finite field and the vector space has a finite dimension. Thus we have F_q, the unique finite field (up to isomorphism, of course) with q elements. Here q must be a power of a prime (q = pm with p prime). Then any n-dimensional vector space V over F_q will have qn elements. Note that the number of elements in V is also the power of a prime. The primary example of such a space is the coordinate space (F_q)n.