I am trying to teach myself tensor products, and I often see claims that the basis for a type (p,q) tensor is of the form: $e_{i_1}\otimes \ldots e_{i_q}\otimes e^{j_1}\otimes \ldots \otimes e^{j_p}$. I have never seen explicit proofs that this is a basis by showing linear independence, so I am trying to do this on my own in a simple and explicit case for a tensor of type (1,1).
I am assuming my vector space V is 2 dimensional with basis $e_1,e_2$, and $V^*$ with basis $e^1, e^2$, such that $e^i(e_j) = \delta ^i_j$. Then for a (1,1) tensor I wish to show the linear independence of:
$$ T_{11} (e_1\otimes e^1) + T_{12} (e_1\otimes e^2) + T_{21} (e_2\otimes e^1) + T_{22} (e_2\otimes e^2) = 0$$
But I am unsure how to show this to be the case.