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 Fq, 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 Fq 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 (Fq)n.
Read more about this topic: Examples Of Vector Spaces
Famous quotes containing the words finite and/or spaces:
“All finite things reveal infinitude:”
—Theodore Roethke (1908–1963)
“When I consider the short duration of my life, swallowed up in the eternity before and after, the little space which I fill and even can see, engulfed in the infinite immensity of spaces of which I am ignorant and which know me not, I am frightened and am astonished at being here rather than there. For there is no reason why here rather than there, why now rather than then.”
—Blaise Pascal (1623–1662)