Express from one dyad base to another

Question

There is a dyad base $\overrightarrow{e_i}\otimes \overrightarrow{e_j}$ and a dyad base $\overrightarrow{e_i}\otimes \overrightarrow{e^k}$ and I understand that the second base can be expressed through the first like this:$\overrightarrow{e_i}\otimes \overrightarrow{e^k} = g^{kj}\overrightarrow{e_i}\otimes \overrightarrow{e_j}$.But there are questions, how it can be painted in more detail, I look in textbooks and I can't understand how it turned out

Answer 1

First of all, let us talk about the most basic property of the metric tensor, raising and lowering indices. In your case, the index $j$ in the first dyad is raised to the index $k$ in the new basis; $g^{kj}\vec{e_{j}} = \vec{e^{k}}$. Since the tensor product on polyads generally produces scalar components, it is safe to say that the (in this case) inverse metric tensor components lower the index on the dyad components.