I got asked to prove by conjugation $P\cdot A=PAP^{-1}$ that $GL_{n}(\mathbb{F})\curvearrowright \mathbb{F}^{n\times n}$ where $GL_{n}(\mathbb{F})$ is the general liner group.
I'm not sure that I understand what does it mean to prove by conjugation. I know the theorm behind the $\curvearrowright$ operator, but I'm not sure how to connect it with $P\cdot A=PAP^{-1}$.
The theorm I understand: let $G$ be a group and $X$ is a set. Then a (left) group action $\phi$ of $G$ on $X$ is a function:
$$G\times X \to X:(g,x) \to \phi (g,x)$$
In order to show the theorm we should prove identity and compatibility. I just don't understand how.