I'm currently working on a problem in the context of the trivial complex vector bundle $\mathbb CP^n\times \mathbb C^{n+1}$ where at one point I am supposed to verify that the connection $1$-form $\omega$ of its respective connection $\nabla$ is pointwise $\mathbb C$-linear.
The respective connection form is defined as $$\omega = \frac{\sum_{k=1}^n\overline{z}_kdz_k}{1+\Vert z\Vert^2}$$
Now (pointwise) $\omega$ being a $1$-form means we consider $$\omega\colon T_{[1,z]}\mathbb CP^n\to \mathbb C$$ So it should suffice to verify that $$\omega(\lambda X_{[1,z]}) = \lambda\omega(X_{[1,z]}),\quad \lambda \in \mathbb C,\ X_{[1,z]} \in T_{[1,z]}\mathbb CP^n $$
But I am uncertain what $\omega(X_{[1,z]})$ evaluates to. I am still not feeling very comfortable working with these objects, e.g. connection forms etc. Any help is highly appreciated!