Assume $B=\{b_1,...,b_n\}$ is a basis of some vector space $V$ and let $C,D$ be other bases of $V$. $[b_i]_C=[b_i]_D \forall i \rightarrow C=D$?

Question

Assume $B=\{b_1,...,b_n\}$ is a basis of some vector space $V$ and let $C,D$ be other bases of $V$. If $[b_i]_C=[b_i]_D$ for all $1\le i\le n$, does that imply $C=D$? I am trying to prove this claim in order to prove some other statement, but I’m starting to have doubts this is true. I know obviously that for some $v\in V$, $v$ can have the same coordinate tuple with respect to different bases of $V$, but since $B$ is a basis itself and $[b_i]_C=[b_i]_D\forall i$, can I conclude that $C=D$? Thanks in advance.

Answer 1

If every element of $B$ has the same coordinates in both bases, then the same thing holds for the linear combinations of elements of $B$. But every element of the space is such a linear combination.

Now take the first element of $C$. Its coordinates with respect to $C$ are $(1,0,0,\ldots,0)$ and so this is also true with respect to $D$. That is, the first element of $C$ is also the first element of $D$. And the same argument applies to all other elements of $C$.