Tensors

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

Quote:

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!