How can we show a tensor map in to some matrix representation? For instance if we have some tensor $T$ is a $(1,1)$ tensor on $\mathbb{R}^{2n}$ given by:
$$T =\sum_{i=1}^n dx^i \otimes \frac{\partial}{\partial x_{i+n}}$$
Can anyone show me how this tensor will have an explicit matrix representation respect to some basis $\{\frac{\partial}{\partial x_i}\}$ for $i=1,...2n?$
A $(1, 1)$-tensor $T$ on $\mathbb{R}^{2n}$ can be written as
$$T = \sum_{i, j = 1}^{2n}T_{ij}\,dx^i\otimes\frac{\partial}{\partial x^j}.$$
The matrix representation of $T$ is then the $2n \times 2n$ matrix $[T_{ij}]$.
For the tensor in your question, we have
$$T_{ij} = \begin{cases} 1 & 1 \leq i \leq n, j = i + n\\ 0 &\ \text{otherwise}. \end{cases}$$
It follows that the matrix representation of $T$ is
$$\left[ \begin{array}{c|c} 0 & I_n\\ \hline 0 & 0 \end{array}\right].$$