Let $V$ be a vector space over a field $K$ We say that $V$ is finite-dimensional if there exists a finite basis of $V$ Otherwise we call $V$ infinite-dimensional.

It can be shown that every basis of $V$ has the same cardinality. We call this cardinality the dimension of $V$ In particular, if $V$ is finite-dimensional, then every basis of $V$ will consist of a finite set $v_1,\ldots, v_n$ We then call the natural number $n$ the dimension of $V$

Next, let $U\subset V$ a subspace. The dimension of the quotient vector space $V/U$ is called the codimension of $U$ relative to $V$

In circumstances where the choice of field is ambiguous, the dimension of a vector space depends on the choice of field. For example, every complex vector space is also a real vector space, and therefore has a real dimension, double its complex dimension.