Dear Max,
Thank you very much for your help. I will try again.
With best wishes,
Jianrong.
> -----原始邮件-----
> 发件人: "Max Horn" <m...@quendi.de>
> 发送时间: 2013年1月7日 星期一
> 收件人: "李建荣" <l...@lzu.edu.cn>
> 抄送:
> 主题: [SPAM] Re: [GAP Forum] Coefficients in GAP
>
> Dear Jianrong,
>
>
> On 06.01.2013, at 14:52, 李建荣 wrote:
>
> > Dear Forum,
> >
> > I would like to compute coefficients of a vector in a vector space V with
> > respect to some basis B.
> > The command in GAP is Coefficients(B, v);
> >
> > If B is given by a list, then it seems that it doesn't work.
>
> Indeed, because as the documentation of Coefficients states, B must be a
> basis -- and a "basis" here is a special GAP object, which is more than just
> a list of basis vectors.
>
> One of the reasons for this is that for an arbitrary list of vectors, such as
> yours, it is not at all clear whether it actually forms a basis of the space
> spanned by those vectors, i.e. if they are linearly independent.
>
>
> > For example, Coefficients([Basis(B)[1], Basis(B)[2], Basis(B)[3],
> > Basis(B)[4], Basis(B)[5], Basis(B)[6], Basis(B)[7], Basis(B)[8],
> > Basis(B)[9], Basis(A)[2]], Basis(B)[4]*Basis(A)[3]) will have errors. Here
> > B is a subspace of the vector space A. A consists of matrices. The codes
> > are in the end of the email. How should I correct
> > Coefficients([Basis(B)[1], Basis(B)[2], Basis(B)[3], Basis(B)[4],
> > Basis(B)[5], Basis(B)[6], Basis(B)[7], Basis(B)[8], Basis(B)[9],
> > Basis(A)[2]], Basis(B)[4]*Basis(A)[3])? Thank you very much.
>
> This look as if you want to form a new basis by taking a subset of the
> existing Basis(B). Here is a sketch of how this could look like.
>
> # Set F to your base field, e.g. GF(3^2)
> F := ...;
>
> # Take first 9 basis vectors of B plus second basis vector of A
> vecs := Concatenation( Basis(B){[1..9]}, Basis(A){[2]} );
>
> # Compute a basis of the spanned vector space:
> bas := ImmutableBasis( MutableBasis( F, vecs ) );
>
> # Optionally, check whether "vecs" really formed a basis.
> if Length(vecs) <> Length(bas) then
> Error("vectors do not form a basis"
> fi;
>
> # Finally, attempt to rewrite a matrix using this basis
> Coefficients( bas, Basis(B)[4]*Basis(A)[3] )
>
>
> But looking at your example code, and considering your previous question, it
> seems that perhaps all you want to do is to check whether some vector is
> contained in the span of some other vectors. Moreover, your code seems to
> contain a mistake (it sets number:=1 inside the loop over k, but tests it
> only after that loop; hence, only the result of the last loop iteration has
> any effect). Supposing all this, you could try something like the following
> (here I assumed you wanted to set and test "number" outside the k-loop; if
> the correct thing for you problem is to check "number" on each iteration of
> that loop, of course you'll have to move things around accordingly):
>
> for i in complement do
> vecs := Concatenation( Basis(B){[1..9]}, Basis(A){[i]} );
> sub := VectorSpace( F, vecs );
> number := true;
> for k in [1..9] do
> for l in vecs do
> if not Basis(B)[k] * l in sub then
> number:=false;
> fi;
> od;
> od;
> if number=true then Add(new_indices_in_LminusOne, i); fi;
> od;
>
>
> The code can actually be simplified a lot more, e.g. by using ForAny or
> ForAll; and more so if, for example, your space B is actually 9-dimensional.
>
>
> Hope that helps,
> Max
>
>
> >
> > With best wishes,
> > Jianrong.
> >
> >
> >
> > for i in complement do
> > for k in [1..9] do
> > number:=1;
> > for l in [Basis(B)[1], Basis(B)[2], Basis(B)[3], Basis(B)[4], Basis(B)[5],
> > Basis(B)[6], Basis(B)[7], Basis(B)[8], Basis(B)[9], Basis(A)[i]] do
> > if Coefficients([Basis(B)[1], Basis(B)[2], Basis(B)[3], Basis(B)[4],
> > Basis(B)[5], Basis(B)[6], Basis(B)[7], Basis(B)[8], Basis(B)[9],
> > Basis(A)[i]], Basis(B)[k]*l)=fail then number:=0; fi;
> > od;
> > od;
> > if number=1 then Add(new_indices_in_LminusOne, i); fi;
> > od;
> >
> > _______________________________________________
> > Forum mailing list
> > Forum@mail.gap-system.org
> > http://mail.gap-system.org/mailman/listinfo/forum
> >
>
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