I'm trying to answer the following question: Which is the minimum number of projective subspaces of dimension $3$ necessary to generate one of dimension $11$.
My interpretation is the following: we want to know the minimum number $n$ of projective subspaces $X_1 \dots X_n$ such that $dim(X_i) = 3$ for all $i=1, \dots, n$ such that $X_1 + X_2 + \dots + X_n$ has dimension 1. We can suppose we are working in a projective space of dimension $k$ with $k \geq 11$
I think that the minimum number of subspaces is $3$. Any subspace of dimension $11$ can be specified by $12$ points in general position $\{x_0, \dots, x_{11}\}$. If we pick our subspaces as $X_1 = V\{x_0, \dots, x_3\}$, $X_2 = V\{ x_4, \dots, x_7\}$, $X_3 = \{x_8, \dots, x_{11} \}$ then $X_1 + X_2 + X_3$ has dimension $11$. Is this analysis correct?