= Solution
Extend $J$ complex-linearly to the <complexification of a real vector bundle> $T_{\mathbb C}X=TX\otimes_{\mathbb R}\mathbb C$. Since $J^2=-\operatorname{id}$, its $+i$ and $-i$ <eigenbundles> have smooth projections
$$
P^{1,0}=\tfrac12(\operatorname{id}-iJ),\qquad
P^{0,1}=\tfrac12(\operatorname{id}+iJ),\qquad
T_{\mathbb C}X=T^{1,0}X\oplus T^{0,1}X.
$$
Each has complex rank $n$: <complex conjugation> interchanges them, and together they have rank $2n$. These are the <type decomposition of the complexified tangent bundle>. For an arbitrary <almost complex structure>, they are smooth complex <vector bundles>; a <holomorphic vector bundle> structure requires integrability.
When $J$ comes from a <holomorphic atlas>, write a <holomorphic coordinate> as $z_j=x_j+iy_j$. Its <Wirtinger derivatives> are
$$
\frac\partial{\partial z_j}=\frac12\left(\frac\partial{\partial x_j}-i\frac\partial{\partial y_j}\right),\qquad
\frac\partial{\partial\bar z_j}=\frac12\left(\frac\partial{\partial x_j}+i\frac\partial{\partial y_j}\right).
$$
The induced <almost complex structure> has $J\partial_{x_j}=\partial_{y_j}$ and $J\partial_{y_j}=-\partial_{x_j}$. Hence the displayed <vector fields> are respectively $+i$ and $-i$ <eigenvectors>. They are linearly independent, and each collection has $n$ elements. Thus \b[they give local frames for $T^{1,0}X$ and $T^{0,1}X$ respectively], with $T^{1,0}X$ the <holomorphic tangent bundle>. The repeated $T^{0,1}X$ in the first sentence of the printed item must be read as the two complementary eigenbundles.
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