Show if a homomorphism has a non trivial kernel

Question

For Q1(a) I have shown, by the property of distribution for matrices, that 1(a) is a homomorphism.

I have tried to show that it only has a non-trivial kernel but cannot find an example. Would I be correct in saying it only has a trivial kernel and how should I show this for the general case?

Thank you for your help in advance.

Answer 1

Hint $:$ Use rank nullity theorem to show that $\text {nullity} (A) = m - \text {rank} (A) \geq m-n > 0$ and observe that $ker\ (\phi)$ is the null space of $A.$