What is the Inner Product between Tensors in Cartesian System?

[Telemachus]

Hi there. I have this problem, which says: In the cartesian system the tensor T, twice covariant has as components the elements of the matrix:
$$\begin{bmatrix}{1}&{0}&{2}\\{3}&{4}&{1}\\{1}&{3}&{4}\end{bmatrix}$$

If $$A=e_1+2e_2+3e_3$$ find the inner product between both tensors. Indicate the type and order of the resultant tensor.

Well, I don't know how to do this. Which type of tensor is A? I think that could help.
The inner product is defined for tensors of different kinds as:
$$S=u^iv_i$$

The supraindex indicates contravariance and the subindex covariance.

 

[tiny-tim]

Hi Telemachus!

I think they're saying that A is first-order contravariant, so T.A will be T^ijA_j

(btw, not what i'd call a dot product )

 

[Telemachus]

Why not?

Thank you Tim :)