Linear algebra MOC

Span and spanning sets

The span⁡(𝐴) ≤𝑉 given a set of vectors 𝐴 ⊆𝑉 is the smallest possible vector subspace containing the vectors of 𝐴. linalg In this way, span⁡(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.

span⁡𝑆={𝜆1⃗𝐯1+⋯+𝜆𝑛⃗𝐯𝑛:𝜆𝑖∈𝕂,⃗𝐯𝑖∈𝑆}

Note the special case

span⁡(∅)={⃗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.

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Footnotes

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