Definition:Generator of Vector Space
Let $K$ be a division ring.
Let $\mathbf V$ be a vector space over $K$.
Let $S \subseteq \mathbf V$ be a subset of $\mathbf V$.
Some sources refer to a generator for rather than generator of. The two terms mean the same.
It can also be said that $S$ generates $\mathbf V$ (over $K$).
Other terms for $S$ are:
- A generating set of $\mathbf V$ (over $K$)
- A generating system of $\mathbf V$ (over $K$)
Some sources refer to such an $S$ as a set of generators of $\mathbf V$ over $K$ but this terminology is misleading, as it can be interpreted to mean that each of the elements of $S$ is itself a generator of $\mathbf V$ independently of the other elements.
- Results about generators of vector spaces can be found here.