Cardinality of a vector space ($|V|=\max (|K|,{\dim }_{K}V)$)

For a nonzero vector space with an infinite field or infinite basis, cardinality is the displayed maximum, since vectors are finite linear combinations of basis vectors. Over a finite field, infinite underlying size forces infinite dimension. Over a countably infinite field, every positive finite dimension is already countably infinite. The theory of infinite underlying spaces is therefore categorical in every infinite cardinal for a finite field, but only in uncountable cardinals for a countably infinite field.