Let $A$ and $B$ are $n\times n$ matrices and $x \in C^{n}$.
If $Ax = Bx$ for all $x$ then $A = B$. To prove this I have selected $x$ from Euclidean basis B = {$e_{1},e_{2},...,e_{n}$}.
Then $Ae_{i} = Be_{i}$ implies $i^{th}$ column of A = $i^{th}$ column of B for all $1 \leq i \leq n$.
Hence $A = B.$
Is this proof complete?