Let $V$ be a [[vector space]] and let $W$ be a [[subspace]]. If $\operatorname{dim} W=\operatorname{dim} V$, then $V=W$. Any subspace not [[Equality of sets|equal]] to #### Proof We know [[maximal subsets form a basis]] and if that [[basis for vector space|basis]] for $W$ has the same number of [[Element of a Set|elements]] as any basis for $V$, then it would be satisfactory as a basis for $V$. By definition of [[dimension of a vector space]], that is the case.