Span and spanning sets

The $s p a n (𝐴) \leq 𝑉$ given a set of vectors $𝐴 \subseteq 𝑉$ is the smallest possible vector subspace containing the vectors of $𝐴$ . #m/def/linalg In this way, $s p a n (𝐴)$ may be thought of as a completion of $𝐴$ so that it fulfils the requirements of a subspace, by including all (finite) linear combinations of the vectors in $𝐴$ .

s p a n 𝑆 = {𝜆 1 ⃗ 𝐯 1 + \dots + 𝜆 𝑛 ⃗ 𝐯 𝑛 : 𝜆 𝑖 \in 𝕂, ⃗ 𝐯 𝑖 \in 𝑆}

Note the special case

s p a n (\emptyset) = {⃗ 𝟎}

The conceptual right-inverse of span is that of the spanning set: given a subspace $𝑆$ a spanning set is any set of vectors $𝐴$ which span the subspace, i.e. cover the entire subspace with their linear combinations. Note every vector space has a spanning subset — itself. The smallest possible spanning set of a space¹ (called the most efficient), unique up to Linear map, is called the Vector basis.

#state/tidy | #SemBr | #lang/en

which may or may not be a subspace of a larger underlying space. ↩

Span and spanning sets

Footnotes