I was recently reading Linear Algebra Problem Book by Paul Halmos and he talks about two ways to think about the change of basis. So let $T\in \mathcal{L}(V)$ ($V$ being finite dimensional) and let $B$ denote a basis for $V.$ Then we let $$A=[T]_{BB}=[[Tb_1]_B,[Tb_2]_B...[Tb_k]_B].$$ This is the matrix representing the transformation in the bases $B.$ Now we ask ourselves two questions:
- If a vector in $x\in V$ can be written in the basis $\mathcal{B}=\{b_1,b_2,..,b_k\}$ $$x=\sum_{i}\lambda_ib_i$$ and if in another basis $\mathcal{E}=\{e_1,e_2,...,e_k\}$ we have $$x=\sum_{i}\gamma_ie_i.$$ Then how are the coefficients related?
- We can also ask what happens when two different vectors in two different basis have the same coordinates so $$x=\sum_{i}\alpha_ib_i$$ and $$y=\sum_{i}\alpha_ie_i.$$
I am not completely sure as to how these things are truly the same. But I think that there is something profound going on here and I would like to understand that because what I think is that the first question can be thought of in terms of matrices whereas the second question can be thought of in terms of operators. Furthermore, I recently encountered a problem asking me to show that if there exists bases $B$ and $B'$ such that for $S,T\in \mathcal{L}(V)$ $$[S]_B=[T]_{B'}$$ then $\exists U$ invertible such that $$T=USU^{-1}.$$ And I think that this is quite related to this concept so any insight would be much appreciated.
Let $A=(a_{ij})_{1\leqslant i,j\leqslant k}$ be the matrix such that the entries of its $i$th column are the coefficients of $e_i$ in the basis $\{b_1,b_2,..,b_k\}$. Then$$(\gamma_1,\ldots,\gamma_k)=A.(\lambda_1,\ldots,\lambda_k).$$On the other hand, $y=A.x$. My guess is that it is in this sense that Halmos claims that the two things are the same. Note that $A$ is the change of basis matrix from $\{b_1,b_2,\ldots,b_k\}$ to $\{e_1,e_2,\ldots,e_k\}$.
As far as I am concerned, I had never though about the second point of view that you mentioned.