\begin{definition}[Definition:Generated Subspace/Definition 1]
Let $K$ be a division ring.
Let $\mathbf V$ be a vector space over $K$.
Let $S \subseteq \mathbf V$ be a subset of $\mathbf V$.
The '''subspace generated by $S$''' is the intersection of all subspaces of $\mathbf V$ containing $S$.
\end{definition}