If we have $Ax=Bx,\forall x$ , can we derive that $A=B$?

Question

Suppose we have two matrices $A,B\in \mathbb{R}^{m\times n}$, vector $x\in \mathbb{R}^n$. If we have $Ax=Bx$ holds for all $x$. Can we say that $A=B$? Why?

Answer 1

Hint: Consider what the vector $Ax$ tells us about the matrix $A$ if $x$ is a basis vector.

Answer 2

Hint: take $x = (1,0,\ldots,0)$, what does this say about the first columns of $A$ and $B$? Then...

Answer 3

The components of a matrix are determined only by the action they have on the basis vectors. Here, since $A$ and $B$ have the same action on all vectors, they must hence have the same components too