Definition:Generated Subspace

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 $\mathbf V$.

Definition 1

The subspace generated by $S$ is the intersection of all subspaces of $\mathbf V$ containing $S$.

Definition 2

The subspace generated by $S$ is the set of all linear combinations of elements of $S$.

Also see

• Equivalence of Definitions of Generated Subspace
• Definition:Generator of Vector Space