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(B) : [ModTupRngElt] -> ModTupRng
Create a module that has as its basis the terms of Q (rows of a).
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.
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.
Given a sequence Q of elements belonging to the module M, return true if the terms of Q are linearly independent.[Next] [Prev] [Right] [Left] [Up] [Index] [Root]