Show that two modules VA,VB are isomorphic. Let K be a field and define two matrices: A:=(1103) and B:=(1003) We then define a K[X]-module on V=K2 such that K[X]×V→V,(P,v)→P(A)⋅v where P(A)∈K2×2 is achieved by plugging in matrix A into polynomial P and P(A)⋅v is therefore the matrix-vector-multiplication. We call this module VA⋅VB can be constructed similarly. I want to show that for K=Q the modules VA,VB are isomorphic.

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Show that two modules VA,VB are isomorphic.  Let K be a field and define two matrices:  A:=(1103) and B:=(1003)  We then define a K[X]-module on V=K2 such that  K[X]×V→V,(P,v)→P(A)⋅v  where P(A)∈K2×2 is achieved by plugging in matrix A into polynomial P and P(A)⋅v is therefore the matrix-vector-multiplication. We call this module VA⋅VB can be constructed similarly.  I want to show that for K=Q the modules VA,VB are isomorphic.

Bryce Barry · Accepted Answer

Actually one can define ϕ:VB→VA by ϕ(v)=Sv. This map is bijective and clearly satisfies  ϕ(v1+v2)=ϕ(v1)+ϕ(v2).  We have to show now that ϕ(P(X)v)=P(X)ϕ(v). But  ϕ(P(X)v)=ϕ(P(B)v)=S(P(B)v)=(SP(B))v=(P(A)S)v=P(A)(Sv)=P(X)ϕ(v).

Show that two modules V_{A}, V_{B} are isomorphic. Let K be a field and define two matrice

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