Bases

The application of the functions in this section is restricted either to vector spaces or to torsion-free modules over a Euclidean Domain. 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.

Basis(M): ModTupRng -> [ModTupRngElt]

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

Rank(M): ModTupRng -> RngIntElt

The rank of the free \(R\)-module \(M\).

Coordinates(M, u): ModTupRng, ModTupRngElt -> [RngElt]

Given a vector \(u\) belonging to the rank \(r\) free \(R\)-module \(M\), \(R\) an Euclidean Domain, with basis \(u_1, \ldots, u_r\), return a sequence \([a_1, \ldots, a_r]\) giving the coordinates of \(u\) relative to the \(M\)-basis: \(u = a_1*u_1 + \cdots +a_r*u_r\).