$M$ is a faithful $A$-module.

Question

Let $A = M_2(\mathbb{C})$ and let $M = \mathbb{C}^2$ with its usual $A$-module structure. How do I show that $M$ is a faithful $A$-module? I understand that this amounts to showing that its annihilator is just the zero ideal, but I am not sure what do from there. Any help would be appreciated, thanks.

Answer 1

So the question has an answer:

Suppose $M$ is a matrix whose first column is somewhere nonzero. Then if $v=(1,0)^t$ then $Mv\neq 0$ because it is equal to the first column of $M$. A similar argument works if the second column is nonzero with the vector $(0,1)^t$. Thus no nonzero matrix annihilates the module.