Is every independent subset $S$ of a vector subspace $W$ such that $\operatorname{card}(S)=\dim(W)$ a basis for $W$?

Question

I was reading a theorem that states that if $W$ is a subspace of a finite-dimensional vector space then every independent subset $S \subset W$ can be extended to form a basis of $W$ by repeatedly adding vectors in way such that the resulting set continues to be linearly independent. But, can´t it be the case that we arrive by that process to a independent subset $S^*\subset W$ with $\operatorname{card}(S^*)=\dim(W)$ that is not a basis of W? I suspect that the answer is no, in that case,I would be grateful if you gave me some hints so I could write a proof