Let $V$ be a [[vector space]] and suppose that one [[basis for vector space|basis]] has $n$ [[Element of a Set|elements]], and another basis has $m$ elements. Then $m=n$. #### Proof Since [[larger sets of vectors than the maximal are dependent]], if one of the two, $n$ or $m$, is larger than the other, the larger set is one of [[linearly dependent vectors]], which would mean, while possibly generating $V$, could not be considered a [[basis for vector space|basis]] of $V$ since the definition states that a basis be a set of [[linearly independent vectors]]