# How to represent tensors in a base? And some questions about indices

I was reading some notes about tensor algebra. And they say that you can imagine vectors like objects independent of the coordinates. And once you choose a base for your space $V$, you can write vectors like $$\vec A= A^\mu \vec e_\mu$$ where $\{ \vec e_\mu\}$ is your base for $V$.

And in a similar way you can write a one-form like $$\widetilde A = A_\mu \widetilde e^\mu$$ where $\{ \widetilde e^\mu\}$ is the base of $V^*$ given by the base of $V$.

And I they also say that you can write tensors like $$R_1= R_\mu \, ^\nu \, _\rho (\widetilde e^\mu \otimes \vec e_\nu \otimes \widetilde e^\rho),$$ and $$R_2 = R^\mu \, _\nu \, _\rho (\vec e_\mu \otimes \widetilde e^\nu \otimes \widetilde e^\rho).$$

In all these examples $\vec A ,\widetilde A, R_1, R_2$ are invariant under change of coordinates.

As far as I understand $\vec A \in V$, $\widetilde A \in V^*$, $R_1 \in V^* \otimes V \otimes V^*$ and $R_2 \in V \otimes V^* \otimes V^*$. But I thought that tensors are elements of the set $V \otimes ...\otimes V \otimes V^* \otimes ... V^*$ in that specific order. Now I am very confused.

Can a tensor can be an element of the tensor product between $V$ and $V^*$ in any order or is there a particular order? Also, what is the importance of the rank, since both $R_1$ and $R_2$ have rank $(1,2)$ but they are from different sets, and have indices in different places, some are contravariant while others are covariant.

And last I understand that you can change contravariant indices for covariant indices with the metric tensor $g$. Does that mean that you can interpret the metric as a function between the tensor product of some copies of $V$ and some of $V^*$. For example $$g_{\rho\pi} R^{\mu~~\pi}_{~\nu} = R^{\mu}_{~\nu\rho}$$ Could you interpret that equation as sending elements of $V \otimes V^* \otimes V$ to elements of $V \otimes V^* \otimes V^*$?

Usually books present the tensor products with $V$ first and $V*$ later for simplicity, but it's perfectly possible to have a tensor that takes any number of vectors and covectors in any order. For example, some people define the Riemann tensor with its upper index last: $R_{abc}^{\quad d}$. This also shows that technically the rank is not enough to identify the set of tensors you're using.