Could you please explain to me how to write a tensor in terms of its components and a basis? Actually, I couldn't understand the form of the tensor space basis [which is a product of the basis vectors and the dual basis vectors: "T = Tαβ ωβ ⊗ eα" where "Tαβ" are the components, "ωβ" are the dual basis vectors (of a dual vector space) and "eα" are basis vector(of a vector space)]
Thank you
The kind of tensor you chose is of type "rank two mixed tensor" and this, correctly written, is $$T=T_{\beta}{}^{\alpha}\ \omega^{\beta}\otimes e_{\alpha},$$ also this formal bi-indexed linear combination can be interpreted as a bilinear map $$T:V\times V^*\to\mathbb R$$ which you can treat by \begin{eqnarray*} T(v,f)&=&T_{\beta}{}^{\alpha}\ \omega^{\beta}\otimes e_{\alpha}(v,f),\\ \\ &=&T_{\beta}{}^{\alpha}\ \omega^{\beta}(v)f(e_{\alpha}),\\ \\ &=&T_{\beta}{}^{\alpha}\ v^{\beta}f_{\alpha}, \end{eqnarray*} since
$\omega^{\beta}(e_{\alpha})=\delta^{\beta}_{\alpha}$,
$\omega^{\beta}(v)=v^{\beta}$,
$f=f_{\sigma}\omega^{\sigma}$ and
$f(e_{\mu})=f_{\mu}$.