I am studying continuum mechanics with an introduction of tensor calculus. First of all I wanna say that this is my very first time i see tensor calculus, so I have a lot of things that are not clear to me, hope you guys can help me with some of said things.
One thing I am quite sure about is that a $(0,2)$-tensor is a bilinear form, i.e. (with the right assumptions) a scalar product. So I thought that it could be represented through a matrix, since there is an isomorphism between $n\times n$ matrices and bilinear form on an $n$-dimensional vector space. Once I find out the associated matrix I know how to use the bilinear form, through the row-column product $x^TAy$, now I wanted to do the same with my $(0,2)$-tensor, but I don't know how to find the associated matrix (assuming one does exist).
Now, another thing I know is that a tensor can be represented through its image on the basis vector, i.e.: $T\in\mathcal{T}^r_s\Rightarrow T=T^{i_1,\dots,i_r}_{j_1,\dots,j_s}\,\mathbf{e}_{i_1}\otimes\dots\otimes\mathbf{e}_{i_r}\otimes\mathbf{e}^{j_1}\otimes\dots\otimes\mathbf{e}^{j_s}$ where $\mathbf{e}_{i_1}$ is a vector of the basis of the vector space considered and $\mathbf{e}^{j_1}$ is a vector of the dual basis. My question is, how this writing can be translated in an appropriate matrix? In the particular case of a $(0,2)$-tensor I have that it can be written as $T=T_{a,b}\,\mathbf{e}^{a}\otimes \mathbf{e}^{b}$, how can I calculate the associated matrix?
Hope my question is clear enough :)