I'm getting a bit stuck on what this is telling me:

Consider a tensor of order 2. Define the 9 cartesian components of T by T_{ij}=e_i.T.e_j.
Firstly, is this restricted to \mathbb{R}^3 since it has 9 cartesian components? In \mathbb{R}^n, would it have n^2 cartesian components?

Secondly, is it true that T_{ij} would be the different entries in a matrix representing T?

Finally, what is e_i.T.e_j? I thought that T was a linear map (well, a tensor) so shouldn't T be defined on how it acts upon a vector, or is this notation that I haven't encountered before?

Thanks for any help!