dimension

$\dim V$ — the number of vectors in any basis of $V$ .

Every basis of a vector space $V$ has the same number of vectors — the invariance of dimension. Because that count never changes, it is a well-defined property of $V$ itself, the dimension, written $\dim V$ . The zero space's only basis is the empty set, so $\dim\{\mathbf 0\}=0$ by convention. For a subspace $W$ of $V$ , $\dim W \le \dim V$ always, with equality forcing $W=V$ .