 Equal-cardinality basis criterion

Let $V$ have finite dimension $n$ and let $S\subseteq V$ have $n$ vectors. Then spanning, linear independence and being a basis are equivalent. From a spanning set, remove redundant vectors until an independent spanning set remains. A removal would give a basis of fewer than $n$ vectors, contradicting dimension invariance. Conversely, if $S$ is independent and a vector lies outside its span, adjoining it gives $n+1$ independent vectors, again impossible. Thus an independent $S$ spans.