Question

Prove that φ : Z ⊕ Z → Z by φ(a, b) = a − b is a homomorphism. Determine the kernel.

Answer 1

To find the kernel of a matrix A is the same as to solve the system AX = 0, and one usually does this by putting A in rref. The matrix A and its rref B have exactly the same kernel. In both cases, the kernel is the set of solutions of the corresponding homogeneous linear equations, AX = 0 or BX = 0.

Now we have  φ((a, b) + (a0 , b0 )) = (a+a0) −(b+b0 ) = φ(a, b) +φ(a 0 , b0 ), and thus φ is a homomorphism.

The kernel of φ is given by {(a, b) | φ(a, b) = 0} = {(a, b) | a − b = 0} = {(a, a) | a ∈ Z}.

Kernel of φ is given by {(a, b) | φ(a, b) = 0} = {(a, b) | a − b = 0} = {(a, a) | a ∈ Z}.