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A.3.11 Kernel of module homomorphisms

Let 190#190, 195#195 be two matrices of size 476#476 and 477#477over the ring 53#53 and consider the corresponding maps

478#478
We want to compute the kernel of the map 479#479This can be done using the modulo command:
480#480
More precisely, the output of modulo(A,B) is a module such that the given generating vectors span the kernel on the right-hand side.

 
  ring r=0,(x,y,z),(c,dp);
  matrix A[2][2]=x,y,z,1;
  matrix B[2][2]=x2,y2,z2,xz;
  print(B);
==> x2,y2,
==> z2,xz 
  def C=modulo(A,B);
  print(C);            // matrix of generators for the kernel
==> yz2-x2, xyz-y2,  x2z-xy, x3-y2z,
==> x2z-xz2,-x2z+y2z,xyz-yz2,0      
  print(A*matrix(C));  // should be in Im(B)
==> x2yz-x3,y3z-xy2, x3z+xy2z-y2z2-x2y,x4-xy2z,
==> yz3-xz2,xyz2-x2z,x2z2-yz2,         x3z-y2z2


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