# Singular Book 2.1.13 -- kernel, image and cokernel of a module homomorphism

In Macaulay2, a Matrix is the same thing as a module homomorphism. The computation of the kernel of a module homomorphism is based on a Groebner basis computation.
 i1 : A = QQ[x,y,z]; i2 : M = matrix{{x,x*y,z},{x^2,x*y*z,y*z}} o2 = | x xy z | | x2 xyz yz | 2 3 o2 : Matrix A <--- A i3 : K = kernel M o3 = image {2} | -y2z+yz2 | {3} | -xz+yz | {2} | x2y-xyz | 3 o3 : A-module, submodule of A
The image and cokernel of a matrix require no computation.
 i4 : I = image M o4 = image | x xy z | | x2 xyz yz | 2 o4 : A-module, submodule of A i5 : N = cokernel M o5 = cokernel | x xy z | | x2 xyz yz | 2 o5 : A-module, quotient of A i6 : P = coimage M o6 = cokernel {2} | -y2z+yz2 | {3} | -xz+yz | {2} | x2y-xyz | 3 o6 : A-module, quotient of A