Prove that if for every $x \in \mathbb{R}^N$ $Ax=Bx$ then $A=B$ How can I quickly prove that if for every $x \in \mathbb{R}^N$
$$Ax=Bx$$ then $A=B$ ? Where $A,B\in \mathbb{R}^{N\times N}$.
Normally I would multiply both sides by inverse of $x$, however vectors have no inverse, so I am not sure how to prove it.

Hint: If $\{e_1,...,e_N\}$ is the standard ordered basis for $\Bbb R^N$, then (for example) $Ae_j$ is the $j$th column of $A$.
