Δ[p_,q_]:=Factor[((((t^(p q))-1)(t-1))/(((t^p)-1)((t^q)-1)))]

Δ[7,3]

u=Table[0,n,n];
Feed[v_]:=Module[{i,vp},
i=1;While[i<=n∧v[[i]]==0,++i];
If[i<=n,
vp=(v/(v[[i]]));
If[u[[i]]===Table[0,n],u[[i]]=vp,Feed[vp-u[[i]]]]]
]

Feed[{0,1,1}]; u

Feed[{1,0,1}]; u

Feed[{1,1,0}]; u

n=10;
mat=Table[(((j+(i^2)-i))^3),{i,n},{j,n}];
mat//MatrixForm

u=Table[0,n,n];
Feed[v_]:=Module[{i,vp},
i=1;While[i<=n∧v[[i]]==0,++i];
If[i<=n,
vp=v/v[[i]];
If[u[[i]]===Table[0,n],u[[i]]=vp,Feed[vp-u[[i]]]]]
]

Feed[mat[[1]]]; MatrixForm[u]

Feed[mat[[2]]]; MatrixForm[u]

Feed[mat[[3]]]; MatrixForm[u]

Feed[mat[[4]]]; MatrixForm[u]

Feed[mat[[5]]]; MatrixForm[u]

NoFeed[v_]:=Catch[Module[{i,vp},
i=1;While[i<=n∧v[[i]]==0,++i];
If[i<=n,
vp=(v/(v[[i]]));
If[u[[i]]===Table[0,n],Throw[NotInSpan],NoFeed[vp-u[[i]]]],
Throw[InSpan]]
]]

NoFeed[Table[(((j+1))^4)-(j^4),{j,n}]]

NoFeed[Table[(((j+1))^5)-(j^5),{j,n}]]