Spanning Set (Generator)

Definition: Spanning Set (Generator)

Let $(V,F,+,\cdot )$ be a vector space.

A set of vectors $S=\{{\mathbf{v}}_1,{\mathbf{v}}_2,\cdots \}$ is called a spanning set or a generator of $V$ if their span is equal to $V$, i.e. every vector in $V$ can be expressed as a linear combination of vectors from $\{{\mathbf{v}}_1,{\mathbf{v}}_2,\cdots \}$:

$$\mathrm{span}({\mathbf{v}}_1,{\mathbf{v}}_2,\cdots )=V$$

Definition: Minimality

A spanning set $S$ is minimal if there is no vector ${\mathbf{v}}_i\in S$ such that $S\setminus \{{\mathbf{v}}_i\}$ is still a spanning set.

Definition: Finitely Generated Vector Space

A vector space is finitely generated if it has a finite spanning set.