If $\mathbb{k}$ is a division ring then $\mathbb{k}^n$ is a simple $M_n(\mathbb{k})$ module

Question

Problem: If $\mathbb{k}$ is a division ring then $\mathbb{k}^n$ is a simple $M_n(\mathbb{k})$ module

I'm lost on this problem, the hint is to use linear algebra but i dont see how it helps.

Answer 1

Hint : First try to see that it's enough to show "for any nonzero vector $v$, there is $M\in M_n(\mathbb k)$ with $Mv = (1,0,...,0)$".

Then try to prove that, using a nonzero coordinate of $v$ and the fact that $\mathbb k $ is a division ring.