Spanvärden

Spanvärden is a term used in Swedish to refer to the elements that belong to the span of a given set of vectors in a vector space. If S = {v1, v2, ..., vk} is a subset of a vector space V over a field F, the span of S, denoted span(S), is the set of all finite linear combinations a1v1 + a2v2 + ... + akvk, where ai ∈ F. The span is a subspace of V and consists of exactly those vectors that can be formed from S through linear combination.

Key properties include that span(S) is the smallest subspace of V containing S, it always contains the zero vector, and it is closed under vector addition and scalar multiplication. If span(S) equals a subspace W, then W is said to be spanned by S. The concept extends to infinite sets as well, though practical computations often focus on finite generating sets.

Examples help illustrate span values. In R^2, span{(1,0), (0,1)} equals all of R^2, while span{(1,2)} is the line through the origin consisting of all multiples of (1,2). If S is empty, span(S) = {0}. A set that spans a subspace is called a spanning set; a basis is a spanning set that is also linearly independent, and the dimension of the subspace equals the size of its basis.

See also: linear span, subspace, basis, linear independence, span in linear algebra, matrix rank.