When trying to deeply visualize the meaning of the entries of a basis transformation matrix, I realized that a (forward) change of basis matrix can be written in a tensor form like this:
$$Q=Q^i_j \; \vec{e_i}\otimes\tilde{\epsilon}^j$$
For completeness sake, the backwards transformation matrix could be written like this:
$$P=P^i_j \; \vec{\tilde{e}_i}\otimes\epsilon^j$$
Where the tildes denote the new basis, and $\epsilon$ stands for the respective dual bases.
Strangely enough, but appropriately, the basis for the matrix is a tensor product of a basis vector and covector (but of different bases!) In this way, it seems like an isomorphism can be made between linear maps and basis transformations, but that they aren't strictly one and the same (i.e. basis transformations are not linear maps, and certainly linear maps are not basis transformations).
Does this seem correct? Or is this an abuse of notation?
Wheeler has popularized the 'northwest-southeast' notation of linear basis transforms in Einstein notation in differential geometry, reduced to the tangent and cotangent basis in two systems.
$$dx^{i'} = \Lambda^{i'}_{\ \ k} dx^k$$
$$ e_k =\partial_k = \Lambda^{i'}_{\ \ k} \ e_{i'} = \Lambda^{i'}_{\ \ k} \ \partial_{i'}$$
$${\Lambda^{-1}}^{i'}_{\ \ k} = \Lambda^k_{\ \ i'}$$
As an intertwiner between to two different spaces, the Jacobi matrix can be viewed as the coeffient matrix of
$$ \Lambda = \Lambda^{i'}_{ \ \ k} \ \ e_{i'} \otimes dx^k $$ in the direct sum of all four tangent spaces, replacing the Einstein index summation convention by contraction of mixed tensor products explicitely.