So I'm reading this paper by Donaldson on contructing symplectic submanifolds, https://projecteuclid.org/euclid.jdg/1214459407
In section 2, he says the following: On ${\bf C}^n$, we have the standard complex structure that gives rise to the decomposition $Hom_{\bf R}({\bf C}^n, {\bf C}) = \Lambda^{1, 0} \oplus \Lambda^{0, 1} $, and he claims that any other complex structure $J$ on ${\bf C}^n$ can be specified from the standard complex structure and a linear map $\mu:\Lambda^{1, 0} \to \Lambda^{0, 1} $ by setting the $J$-linear forms $\Lambda^{1, 0}_J$ to be those of the shape $\phi + \mu(\phi)$. (i.e $\Lambda^{1, 0}_J$ is the graph of $\mu$)
I don't see why this is true, what if $J=-i$, then we have that $\Lambda^{1, 0}_J= \Lambda^{0, 1}$, and clearly it cannot be given as the graph of a function from $\Lambda^{1, 0}$ to $\Lambda^{0, 1}$