linearindB

LinearindB is a term encountered in some texts and software libraries to denote a linear independent basis of a subspace, often derived from a given set of vectors. In this usage, linearindB refers to an ordered subset B = {b1, ..., br} of a vector space V such that the vectors in B are linearly independent and their span equals the subspace of interest (or, more specifically, the span of the original generating set).

Construction and properties

A linearindB can be obtained by standard algorithms in linear algebra, such as Gaussian elimination, row reduction to row echelon form, or incremental methods that add vectors to B only if they are not expressible as a linear combination of the current members. The resulting B is a basis for the subspace, so its size r equals the subspace dimension. The choice of linearindB is typically not unique; different valid sequences of added vectors can yield different bases for the same subspace. The vectors within a linearindB are linearly independent, and every vector in the subspace can be expressed as a linear combination of the basis elements.

Examples

In R^3, the standard basis e1 = (1,0,0), e2 = (0,1,0), and e3 = (0,0,1) forms a linearindB for the whole space. For the x-y plane, a linearindB can be {(1,0,0), (0,1,0)}. More generally, any maximal linearly independent subset of a generating set is a linearindB.

Applications

LinearindB is used in solving linear systems, performing change of basis, determining subspace dimensions, and implementing linear transformations in computational linear algebra.

See also

linear independence, basis, vector space, span, rank, Gaussian elimination.