Should multiplying a MatrixObj with a list work? Always? Never? Current state: sometimes, at random #6635

Yesterday @GalaxyCo asked me how to multiply MatrixObj objects with lists of coefficients. This quickly turned into "should this work"? Maybe one should require using a "compatible" VectorObj

This raised the question how to even get such a VectorObj (which lead to issue #6634).

But also it lead to experiments which uncovered a lot of irregularities. Sometimes it fails:

gap> F:=GF(2);;
gap> m:=Matrix(IsGenericMatrixRep, F,  IdentityMat(3, F));;
gap> v:=[1,2,3]*One(F);;
gap> m*v;
Error, no method found! For debugging hints type ?Recovery from NoMethodFound

But it weirdly works if v is a compressed vector (maybe because those are VectorObj, even if they are not declared as "compatible" here):

gap> ConvertToVectorRep(v);
2
gap> v;
<a GF2 vector of length 3>
gap> m*v;
<a GF2 vector of length 3>

As a hack workearound I inserted this:

InstallOtherMethod( Vector,
    [ IsList and IsFFECollection, IsList and IsFFECollection ],
    { v, example } -> v );

which makes both work. Before you complain too much about it, please consider that this is already in the library:

# The following method is used for example
# in code dealing with elements of 'AlgebraicExtension's.
# (Note that we want to return a vector with the same mutability status
# as the given vector, and we need not copy it if it is mutable.)
InstallOtherMethod( Vector,
    [ IsList and IsCyclotomicCollection, IsList and IsCyclotomicCollection ],
    { v, example } -> v );

Anyway what about characteristic 0? Well, it already works for integers:

gap> F:=Integers;;
gap> m:=Matrix(IsGenericMatrixRep, F,  IdentityMat(3, F));;
gap> v:=[1,2,3]*One(F);;
gap> m*v;
[ 1, 2, 3 ]

However, it still fails for e.g. rationals:

gap> F:=Rationals;;
gap> m:=Matrix(IsGenericMatrixRep, F,  IdentityMat(3, F));;
gap> v:=[1,2,3]*One(F);;
gap> m*v;
Error, <M> and <v> are not compatible

My guess is that this is because GAP "guesses" the basedomain for the vector as Integers (TODO: we really should improve the error message there to give out this information).

Indeed, we can confirm my theory:

gap> m*(v/2);
[ 1/2, 1, 3/2 ]