Definition of a Module

Construction of Modules of \(n\)-tuples

RSpace(R, n): Rng, RngIntElt -> ModTupRng
RModule(R, n): Rng, RngIntElt -> ModRng: Given a ring \(R\) and a non-negative integer \(n\), create the free right \(R\)-module \(R^{(n)}\), consisting of all \(n\)-tuples over \(R\). The module is created with the standard basis, \(e_1, \ldots, e_n\), where \(e_i\) \((i = 1, \ldots, n)\) is the vector containing a \(1\) in the \(i\)-th position and zeros elsewhere. The function RModule creates a module in reduced mode while RSpace creates a module in embedded mode.

RSpace(R, n, F): Rng, RngIntElt, Mtrx -> ModTupRng: Given a ring \(R\), a non-negative integer \(n\), and a square \(n \times n\) symmetric matrix \(F\), create the free right \(R\)-module \(R^{(n)}\) (in embedded form), with inner product matrix \(F\). This is the same as RSpace(R, n), except that the functions Norm and InnerProduct (see below) will be with respect to the inner product matrix \(F\).

Example: Module of 6-Tuples over the Integers (ex-d42492)

We construct the module consisting of \(6\)-tuples over the integers.

> Z := IntegerRing();
> M := RModule(Z, 6);
> M;
RModule M of dimension 6 with base ring Integer Ring

RMatrixSpace(R, m, n): Rng, RngIntElt, RngIntElt -> ModMatRng: The module comprising all \(m \times n\) matrices over the ring \(R\).

RModuleWithBasis(Q): [ModFldElt] -> ModFld
RSpaceWithBasis(Q): [ModTupRngElt] -> ModTupRng
RSpaceWithBasis(a): AlgMatElt -> ModTupRng
RSpaceWithBasis(a): ModMatRngElt -> ModTupRng: Given a sequence \(Q\) (or matrix \(a\)) of \(k\) independent vectors each lying in a module \(M\), construct the submodule of \(M\) 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.

RMatrixSpaceWithBasis(Q): [ModTupRngElt] -> ModMatRng: The module of \(m \times n\) matrices whose basis is given by the linearly independent matrices of the sequence \(Q\).