I'm trying to study tensors from several textbooks. I would like it if someone could confirm my understanding of a particular easy example. It is from Itskov, Tensor Algebra and Tensor Analysis for Engineers, 3rd edition.
$\mathbf{e}_i$ is an orthonormal basis, and $\mathbf{g}_i$ is some other basis. $\mathbf{e}^i$ and $\mathbf{g}^i$ are the respective dual bases.
Itskov expresses the "primal" bases in terms of each other (eq. 1.16):
$\mathbf{e}_i = \alpha_i^j \, \mathbf{g}_j \quad\text{and}\quad \mathbf{g}_i = \beta_i^j \,\mathbf{e}_j$
Ok, no problem so far.
Itskov then says ``Let further'' (eq. 1.19):
$\mathbf{g}^i = \alpha^i_{\,j} \, \mathbf{e}^j$
My question: I believe that this $\alpha^i_{\,j}$ is a different thing than the $\alpha_i^j$ in 1.16; I think I should regard these as matrix inverses of each other. However the notation almost suggests that they could be the same numbers, just indexed differently.
One possible little clue, the subscript $j$ in 1.19 is displaced slightly (which I tried to reproduce here).
Using your notation, suppose $g^{i} = \sum_{j}\beta^{i}_{j}e^{j}$.
Then $$\beta^{i}_{j} = g^{i}(e_{j}) = g^{i}(\sum_{k}\alpha^{k}_{j}g_{k}) = \sum_{k}\alpha^{k}_{j}\cdot g^{i}(g_{k}) = \alpha_{j}^{i}$$
So $\beta^{i}_{j} = \alpha_{j}^{i}$