vectoriset

Vectoriset, in Finnish usage, refers to a collection of vectors drawn from a common vector space. The elements are vectors, objects that can be added and scaled by field elements, typically real numbers. A vectoriset may be finite or infinite and may reside in any dimension.

The central concept is the span. The span of a vectoriset S consists of all finite linear combinations of vectors in S. If the span equals a particular subspace, S is said to generate that subspace. A vectoriset is linearly independent if no nontrivial linear combination of its vectors equals the zero vector; otherwise it is linearly dependent.

A vectoriset that is both linearly independent and spans its subspace is a basis for that subspace. The number of vectors in a basis is the dimension of the subspace. In a finite-dimensional space, every spanning vectoriset contains a basis, and every linearly independent vectoriset can be extended to a basis.

Matrix view: A vectoriset consisting of n vectors in F^m can be arranged as an m×n matrix with the vectors as columns. The rank of this matrix equals the dimension of the span of the vectoriset.

Applications: Vector sets are fundamental in solving linear systems, describing subspaces, and representing data as coordinate vectors in machine learning or analytics.

See also: linear algebra, basis, span, linear independence, subspace, matrices.