dimension (vector space)

Let $V$ be a vector space over a field $K$. We say that $V$ isfinite-dimensional if there exists a finite basis of $V$. Otherwise wecall $V$ 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 $v_1,\mathrm{\dots },v_n$. We then call the natural number $n$ the dimension of $V$.

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

In circumstances where the choice of field is ambiguous, thedimension of a vector space depends on the choice of field. Forexample, every complex vector space is also a real vector space, andtherefore has a real dimension, double its complex dimension.