= Cardinality of a vector space
{title2=$|V|=\max(|K|,\dim_KV)$}
For a nonzero <vector space> with an infinite <field> or infinite <vector space basis>[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 of a vector space>[dimension]. Over a countably infinite <field>, every positive finite <dimension of a vector space>[dimension] is already countably infinite. The <first-order theory>[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>.
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