Subsections

• Bases for Free Modules

This subsection is concerned with the construction of bases for R-modules. Consequently, the application of these functions is restricted either to vector spaces or to torsion-free modules over a Euclidean Domain. If these operations are applied to a R[G]-module, they are considered to act on the underlying R-module.

For a full description of the basis functions for a module defined over a field, the reader is referred to the chapter on vector spaces.

RModuleWithBasis(Q) : [ModTupRngElt] -> ModTupRng

RSpaceWithBasis(Q) : [ModTupRngElt] -> ModTupRng

RModuleWithBasis(a) : AlgMatElt -> ModTupRng

RSpaceWithBasis(a) : AlgMatElt -> ModTupRng

RModuleWithBasis(a) : ModMatElt -> ModTupRng

RSpaceWithBasis(a) : ModMatElt -> ModTupRng

Given a sequence Q (or matrix a) of k independent vectors each lying in a module V, construct the submodule of V of dimension k whose basis is Q (or the rows of a). The basis is echelonized internally but all functions which depend on the basis of the space (e.g. Coordinates) will use the given basis.

The current basis for the free R-module M, R an ED, returned as a sequence of module elements.

The i-th basis element for the module M.

The current basis for the free R-module M, R an ED, returned as the rows of a matrix belonging to the matrix bimodule R^((m x n)), where m is the dimension of M and n is the over-dimension of M.

The rank of the free R-module M.

Given a vector v belonging to the rank r free R-module M, R an ED, with basis u_1, ..., u_r, return a sequence [a_1, ..., a_r] giving the coordinates of u relative to the M-basis: u = a_1 * u_1 + ... + a_r * u_r.

Given a sequence Q containing r linearly independent vectors belonging to the module M, add sufficient vectors to Q so that the extended set forms a basis for M. The basis is returned in the form of a sequence T such that T[i] = Q[i], i = 1, ... r.

ExtendBasis(N, M) : ModTupRng, ModTupRng -> [ModTupRngElt]

Given a rank r submodule N of the module M, return a basis for M in the form of a sequence T of elements such that the first r terms correspond to the given basis vectors for N.

Given a set S of elements belonging to the module M, return true if the elements of S are linearly independent.

IsIndependent(Q) : [ ModTupRngElt ] -> BoolElt

Given a sequence Q of elements belonging to the module M, return true if the terms of Q are linearly independent.