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 $\displaystyle T_{ij}=e_i.T.e_j$.
Firstly, is this restricted to $\displaystyle \mathbb{R}^3$ since it has 9 cartesian components? In $\displaystyle \mathbb{R}^n$, would it have $\displaystyle n^2$ cartesian components?

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

Finally, what is $\displaystyle e_i.T.e_j$? 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!