So Im trying to explicitly calculate the dual basis for a given basis of $Gl(n, \mathbb{R})$ with $x = (x^i_j)$ and $det(x) \ne 0$. I think this should be rather easy but i cant get to a result that seems to fulfill the $\delta^i_j$ criteria.
My given basis is: $(X_a)_x = x^i_ja^j_k\frac{\partial}{\partial x^i_k}$ with $a = \left[\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{array}\right]$ and $\alpha= \left[\begin{array}{cc} 1 & 0 \\ 0 & 0 \\ \end{array}\right]$ , $\beta = \left[\begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array}\right]$, $\gamma= \left[\begin{array}{cc} 0 & 0 \\ 1 & 0 \\ \end{array}\right]$, $\delta= \left[\begin{array}{cc} 0 & 0 \\ 0 & 1 \\ \end{array}\right]\\$.
With this we define our basis as: $x_\alpha = x^i_1\frac{\partial}{\partial x^i_1},x_\beta = x^i_2\frac{\partial}{\partial x^i_1},x_\gamma = x^i_1\frac{\partial}{\partial x^i_2},x_\delta = x^i_2\frac{\partial}{\partial x^i_2}$.
I now want to find the dual basis $\Theta^k$ with $k = [\alpha, \beta, \gamma, \delta]$ so that $<\Theta^{k}, X_{i}> = \delta^k_i$ for $i,k = [\alpha, \beta, \gamma, \delta]$.
If somebody could even just point me in the right direction it would be much appreciated. :) Thanks in advance!