Why is the Change of Basis map unique? I've been looking all over, but I haven't found anything satisfactory.
We've been shown in class by a commutative diagram that, given an $n$-dimensional vector space $V$ over a field, $\mathbb{F}$, and bases, $\mathcal{B}=\{v_1,...,v_n\}$ and $\mathcal{C}=\{u_1,...,u_n\}$, the coordinate maps $[]_{\mathcal{B}}:V\rightarrow \mathbb{F}^n$ and $[]_{\mathcal{C}}:V\rightarrow \mathbb{F}^n$ give rise to a unique map $P=[]_{\mathcal{B}}\circ []^{-1}_{\mathcal{C}}:\mathbb{F}^n\rightarrow \mathbb{F}^n$, which is our change of basis matrix.
But I am having a lot of trouble proving that $P$ is unique. Can anyone enlighten me as to why this is necessarily true?

Remember that any linear map on any linear space $\;V\;$ is uniquely and completely determined once we know its action on any basis of $\;V\;$ ...and that's all.
If you want to do this proof, suppose there's another map $\;Q:V\to V\;$ s.t. it coincides on "the old basis" $\;\mathcal B\;$ with $\;P:\;\; Qv_i=Pv_i\;\;\forall\,i=1,2,...,n\;$ , then (using linearity of the maps),  for any
$$v=\sum_{k=1}^n a_iv_i\in V\;,\;\;Qv=\sum_{k=1}^na_iQv_i=\sum_{k=1}^n a_iPv_i=Pv$$
so $\;Q\equiv P\;$.
