If $A, B ∈ M_{m×n}$ and $B\vec w = A\vec w$ for any $\vec w ∈ R^n$, then show that $B = A$.
I can prove this by using standar vector as the $\vec w$ as written below. My question is, is this way of proving acceptable? Do I have to use the general vector $\vec w$ instead of a representative vector?
My attempt:
To prove $B=A$, I choose $\vec{e_1}$, $\vec{e_2}$, ..., $\vec{e_n}$. Let $\vec c = B\vec w$ and $\vec d = A\vec w$.
For $\vec {e_1} = \vec w$, $\vec c = B\vec {e_1} = 1\cdot \vec {b_1} + 0\cdot \vec{b_2} + ...+ 0\cdot \vec{b_n} = \vec{b_1}$ and $\vec d = A\vec {e_1} = 1\cdot \vec {a_1} + 0\cdot \vec{a_2} + ...+ 0\cdot \vec{a_n} = \vec{a_1}$. Thus, if $\vec c= \vec d$ then $\vec {b_1} = \vec {a_1}$.
If we continue this for all column in B and A, we can get,
$\forall i=\{1,...,n\}$, let $\vec w = \vec {e_i},$
$c = B\vec {e_i} = ...+ 1\cdot \vec {b_i} + ...= \vec{b_i}$ and $\vec d = A\vec {e_i} = ...+1\cdot \vec {a_i} +...= \vec{a_i}$
Thus, if $B\vec w = A\vec w \iff \vec c = \vec d \Rightarrow \vec{b_i}= \vec{a_i} \iff B=A$!