Definition:Generator of Vector Space

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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 $\mathbf V$.


$S$ is a generator of $\mathbf V$ if and only if every element of $\mathbf V$ is a linear combination of elements of $S$.


Generated Subspace

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 known as

A generator of a vector space is also known as a spanning set.

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.


Also see

  • Results about generators of vector spaces can be found here.


Sources