Spanning Sets

$span {| v_{1} ⟩, \dots, | v_{2} ⟩} = {| v ⟩ : | v ⟩ = \sum_{i}^{n} c_{i} | v_{i} ⟩, c_{i} \in F}$ = {all possible linear combinations of the $| v_{i} ⟩$ }

Definition

The set of linear combinations of $| v_{1} ⟩, | v_{2} ⟩, \dots, | v_{n} ⟩$ is called the span of $| v_{1} ⟩, | v_{2} ⟩, \dots, | v_{n} ⟩$ . The span of $| v_{1} ⟩, | v_{2} ⟩, \dots, | v_{n} ⟩$ is denoted by $span | v_{1} ⟩, | v_{2} ⟩, \dots, | v_{n} ⟩$ .

Examples

Unbounded spaces: span{line segment} = line. span{two non-collinear line segments} = plane. Continuous Function Spaces.

Bounded spaces: Integer rings.