In the definition of a holomorphic line bundle of rank $n$, it is required that the transition maps $g_{ij}:U_i\cap U_j\to\text{GL}_n(\Bbb C)$ are holomorphic.
What does this actually mean? Are we seeing $\text{GL}_n(\Bbb C)$ as a complex manifold? Otherwise I can't even make sense of this being continuous? Is it a problem that these may have highly different dimensions?
I am mainly thinking about Riemann surfaces.