$\mathbf{Problem}$: Let $Gr_{\mathbb{C}} (1,2)$ be the complex Grassmannian manifold that consists of all the complex lines going through the origin in $\mathbb{C}^2$, and $Gr_{\mathbb{R}} (2,4)$ be all the 2-dimensional linear subspaces of $\mathbb{R}^4$. Show that the function $f: Gr_{\mathbb{C}}(1,2) \to Gr_{\mathbb{R}}(2,4)$ is smooth, where f maps a complex line to itself regarded as a 2-dimensional linear subspace of $\mathbb{R}^4$.
$\mathbf{Attempt}$:
1) $\mathbf{Setup}$: Proving the map $f$ smooth is to prove that $\psi_j \circ f \circ \phi^{-1}_i$ is smooth when the composition makes sense. The charts I am using for the Grassmannians are the standard ones: for $Gr_{\mathbb{C}} (1,2)$, define $I = \{1,2\}$, $I_i = \{i\}$, $\mathbb{C}^{I_i} = \{(z_1, z_2): z_j = 0, \forall j \in I - I_i \}$, and $U_{I_i} = \{E \subset \mathbb{C}^2: E \cap \mathbb{C}^{I - I_i} = \{0\} \}$. Define the function $\phi_i: U_{I_i} \to Hom(\mathbb{C}^{I_i}, \mathbb{C}^{I - I_i})$ by $\phi_i(E) = l_E$, the linear map from $\mathbb{C}^{I_i}$ to $\mathbb{C}^{I - I_i}$ such that $E = \{z + l_e(z): z \in \mathbb{C}^{I_i}\}$, i.e., E is the graph of it. Similar definition holds for $Gr_{\mathbb{R}} (2,4)$.
2) Geometrically the problem seems straightforward as I am not really changing anything about the line itself, but I am stuck in proving smoothness of the composition of the functions. In class we defined the smoothness of a function between manifolds by showing $\psi_j \circ f \circ \phi^{-1}_i$ is a smooth function mapping from a Euclidean space to a Euclidean space, which in our case is not exactly so, since $\psi_j \circ f \circ \phi^{-1}_i: Hom(\mathbb{C}^{I_i}, \mathbb{C}^{I - I_i}) \to Hom(\mathbb{R}^{J_j}, \mathbb{R}^{J - J_j})$, so we seem to be mapping linear functions to linear functions. I know in general $Hom(\mathbb{R}^{I'}, \mathbb{R}^{I - I'})$ is isomorphic to $\mathbb{R}^{k(n-k)}$ for $|I'| = k, |I| = n$ (similar case for $Hom(\mathbb{C}^{I'}, \mathbb{C}^{I - I'})$), but how does this work in our context exactly?
3) Another question I have is of more general context. For functions that maps between Euclidean spaces, if the functions is smooth over the Euclidean space, then it must be smooth mapping between the manifolds right?
Any help is appreciated!