[GAP Forum] free module

[Max Horn]

Dear Alper,

> On 04 Oct 2016, at 11:31, Alper Odabaş <aodabas at ogu.edu.tr> wrote:
> 
> Dear forum,
> 
> 
> 
> By linear algebra, the choice of an (ordered) basis for a free module of
> finite rank m yields an isomorphism to Z^{1 x m}, the module whose entries
> are row matrices with m columns.
> 
> 
> 
> In GAP, how to get matrices from the group algebra
> GroupRing(GF(2),CyclicGroup(3))

Your question is a bit ambiguous, I'll interpret it as follows: Given a basis B of an algebra A, how can I express an element x as a matrix over that basis?

Answer: Using the command AdjointMatrix. Here is an example:

gap> R:=GroupRing(GF(2),CyclicGroup(3));
<algebra-with-one over GF(2), with 1 generators>
gap> x:= R.1 + R.1^2;
gap> B:=Basis(R);
CanonicalBasis( <algebra-with-one over GF(2), with 1 generators> )
gap> AsList(B); # let's see which basis GAP picked...:
[ (Z(2)^0)*<identity> of ..., (Z(2)^0)*f1, (Z(2)^0)*f1^2 ]
gap> mat:=AdjointMatrix(Basis(R), x);
[ [ 0*Z(2), Z(2)^0, Z(2)^0 ], [ Z(2)^0, 0*Z(2), Z(2)^0 ], [ Z(2)^0, Z(2)^0, 0*Z(2) ] ]
gap> Display(mat);
 . 1 1
 1 . 1
 1 1 .

Hope that helps,
Max