basisfromS

BasisfromS is a term used in linear algebra to denote a procedure or function that, given a finite set S of vectors in a vector space V over a field F, returns a basis for the subspace spanned by S. The output is a set B such that span(S) = span(B), B is linearly independent, and B is typically chosen as a subset of S, representing a maximal linearly independent subset of S.

Methodically, basisfromS often proceeds by forming a matrix A with the vectors in S as columns. Row-reducing A to reduced row echelon form reveals the pivot columns; the corresponding original vectors form a basis for span(S). Equivalently, an incremental algorithm can be used: start with an empty basis B and, for each vector v in S, append v to B only if v is not in the span of the vectors already in B. The resulting B is a basis for span(S).

Key properties include that the output is not unique; any maximal linearly independent subset of S serves as a basis for span(S). The dimension of span(S) equals the number of basis vectors produced. If an orthogonal or orthonormal basis is desired, one can apply an additional process such as Gram–Schmidt to the basis returned by basisfromS.

Applications of basisfromS include determining the dimension of a subspace, simplifying representations of vectors within the span, solving linear systems, and checking linear independence among a set of vectors. Related concepts are span, linear independence, rank, and the notions of basis and row or column space.