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.

I think it's better to think of what's given as a definition of $A = B$ (since linear maps are isomorphic to matrices by understanding multiplying column matrices as evaluating on vectors).  But otherwise, as others suggested, it's also easy to see that $A = AI = BI = B$, where $I$ is the identity matrix by "cooking up" $I$ for each column.
