Linear algebra MOC

Span and spanning sets

The given a set of vectors is the smallest possible vector subspace containing the vectors of . linalg In this way, may be thought of as a completion of so that it fulfils the requirements of a subspace, by including all (finite) linear combinations of the vectors in .

Note the special case

The conceptual right-inverse of span is that of the spanning set: given a subspace a spanning set is any set of vectors 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. ↩