Let $A$ an open set of $\mathbb{R}^{n}$ and $k \geq 0$ an integer. Consider $\Omega^{k}(A)$ is the vector spaces of all $k$-forms in $A$. Now consider the linear operator $$L: \Omega^{k}(A) \to \Omega^{n-k}$$
characterized by: if $\omega = dx_{i_{1}} \wedge \cdots \wedge dx_{i_{k}}$ then $L(\omega)$ is the $n-k$-form such that $$\omega \wedge L(\omega) = dx_{1} \wedge \cdots \wedge dx_{n}$$
My question is: consider $n = 2$, determine $L(fdx_{1} + gdx_{2})$, where $f$ and $g$ are real functions. How I proceed with the real functions?
From the definition of $L$ you immediately get that $L(dx_1) = dx_2$ while $L(dx_2) = -dx_1$. Since $L$ is linear, you can simply pull the $f$ and $g$ out to get
$$ L(f dx_1 + gdx_2) = fdx_2 - gdx_1. $$