In my notes it's written that, for two basis $B={e_i}$ and $B'=e'_i$, the transformation rule is:
$e'_j = c^i{}_j e_i$
This I can understand, every vector of the $B'$ basis is a linear combination of vectors from the $B$ basis. But when written in matrix notation:
$\begin{pmatrix} e'_1 &...&e'_2 \end{pmatrix} =\begin{pmatrix} e_1&...&e_2 \end{pmatrix} \cdot C$
According to the definitions of $B$ and $B$, every $e_i$ and $e'_i$ is a basis vector. But how would we write this in a real scenario, where we would have to use the components of every vector? Would we just expand every $e_i$ into its column, something like:
$\begin{pmatrix} e'^1_1 & ... &e'^1_n \\ ...& ... & ...\\ e'^m_1&... & e'^m_n \end{pmatrix} =\begin{pmatrix} e^1_1 &... &e^1_n \\ ...&... &... \\ e^m_1& ... &e^m_n \end{pmatrix} \cdot C$
But if you wrote it this way, then every $e'_i$ wouldn't be a linear combination of each $e_i$, every component would be a linear combination of the components of $e_i$, which isn't the same.
So, how would you do it? I suppose an answer is that you can't do it all at once, and that you can only do it with each basis vector at a time. Is that the case?