Av = Bv for all v implies A = B? [closed]

If $A$ and $B$ are two $4\times3$ matrices such that $A\mathbb{v}=B\mathbb{v}$ for all $\mathbb{v}\in\mathbb{R}^3$, then $A=B$?

If it's not true, can you give me an example of it?

Thank you.

Solution 1:

Yes; a matrix is uniquely determined by the linear transformation it defines, and vice versa. One explicit proof is:

$$A=AI = A[e_1 \dots e_n] = [Ae_1 \dots Ae_n] \\ B=BI=B[e_1 \dots e_n] = [Be_1 \dots Be_n]$$

so because $Ae_i=Be_i$ we get $A=B$. Here $e_i$ is the $i$th unit coordinate vector.