Definition:Generated Subspace/Definition 1

Definition
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 $S$.

Also see

 * Intersection of Vector Subspaces is Vector Subspace