I have three vectors, with unit normals, that form the basis of a coordinate system, these are
\begin{align} \boldsymbol{a}_{1} &= [a_{11},\,a_{12},\,0] \,, \\ \boldsymbol{a}_{2} &= [a_{21},\,a_{22},\,0] \,, \text{and} \\ \boldsymbol{a}_{3} &= [0,\,0,\,1] \,. \end{align}
How do I determine the basis and basis vectors $\boldsymbol{a}^{1},\,\boldsymbol{a}^{2},\,\boldsymbol{a}^{3}$ of the adjoint space which is dual to the original space?
Further, how do I verify that the spaces are in fact dual?
(PS: I think it should satisfy $\left<\boldsymbol{a}_{i},\,\boldsymbol{a}^{j}\right>=\delta_{i}^{j}$ but I am not certain.)
Hint: Do an Ansatz like $$\boldsymbol{a}^1=P\varepsilon^1+Q\varepsilon^2+R\varepsilon^3,$$ where $\langle\varepsilon^i,e_j\rangle=\delta^i_j$, to set a linear system of equations $$1=\langle \boldsymbol{a}^1,\boldsymbol{a}_1\rangle$$ $$\qquad\qquad\qquad\qquad\qquad\qquad=\langle P\varepsilon^1+Q\varepsilon^2+R\varepsilon^3,a_{11}e_1+a_{12}e_2\rangle,$$ $$\qquad\qquad\qquad=Pa_{11}+Qa_{12},$$
$$0=\langle \boldsymbol{a}^1,\boldsymbol{a}_2\rangle$$ $$\qquad\qquad\qquad\qquad\qquad\qquad=\langle P\varepsilon^1+Q\varepsilon^2+R\varepsilon^3,a_{21}e_1+a_{22}e_2\rangle,$$ $$\qquad\qquad\qquad=Pa_{21}+Qa_{22},$$
$$0=\langle \boldsymbol{a}^1, \boldsymbol{a}_3\rangle,$$ $$\qquad=R.$$
...etc.