If $f:\mathbb{Z}\times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}$ is homomorphism, show that $f$ is monomorphism.

Question

Here's the complete problem.

If $f:\mathbb{Z}\times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}$ is homomorphism such that $f(0,1)=(-1,5)$ and $f(1,0)=(2,-3)$, show that $f$ is monomorphism.

Need help at least a hint. I have no idea. Thanks for help in advance.

Answer 1

Hint: Prove the kernel is trivial. That is, $f(x,y)=(0,0)\implies (x,y)=(0,0)$.

Further hint: $f(x,y)=f(x(1,0)+y(0,1))=f(x(1,0))+f(y(0,1))=xf(1,0)+yf(0,1)=x(2,-3)+y(-1,5)=(2x-y,-3x+5y)$.