Independent Sets of Right Size are Basis Theorem

Let $V$ be a finite dimensional vector space. If $S$ is a linearly independent set with $|S|=dim(V)$ then $S$ is a basis for $V$

Proof

WTS that $S$ is indep and $|S|=dim(V)$ $⟹$ $v=span(S)$ Suppose $S$ is indep and $|S|=dim(V)$ For sake of contradiction, suppose $V\ne span(S)$ Then, we can extend $S$ to $S^{\prime }=S\cup \overrightarrow{v}$ for $\overrightarrow{v}\notin span(S)$ This gives a linearly independent set in $V$ of size $|S|+1=dim(V)+1>dim(V)$ This contradicts the definition of $dim(V)$. Aka, we get a linearly dependent set that is too big to be in $V$. therefore, $V=span(S)$