Q:

equivalence of two notions of linear subspaces in general position

Let $V_1,\dots,V_n$ be linear subspaces of $k^D$, where $k$ is a field. 
For any subset $S$ of $[n]:=\left\{1,\dots,n\right\}$ define $V_S = \sum_{i \in S} V_i$
and $W_S = \bigcap_{i \in S} V_i$. Define also $d_i = \dim V_i$, $c_i = D - d_i$, $d_S = \dim V_S$, $c_S = D - \dim W_S$.
Question: Are the two following conditions equivalent?
i) $c_S = \min(D, \sum_{i \in S} c_i), \, \, \, \forall S \subset [n]$
ii) $d_S = \min(D, \sum_{i \in S} d_i), \, \, \, \forall S \subset [n]$
Proof or counterexample?
Edit: It can be assumed that $k=\mathbb{R}$ or $k = \mathbb{C}$ if helpful.

A:

These don't appear to be equivalent.
Let $D = 3$ and fix a 1-dimensional subspace $U$ of $\mathbb{R}^3$; then take $V_1, V_2$, and $V_3$ to be distinct 2-dimensional subspaces of $\mathbb{R}^3$ such that $V_1 \cap V_2 \cap V_3 = U$, and so that $V_i + V_j = \mathbb{R}^3$ when $i \neq j$.
Then (ii) holds. (Obvious for 1-element subsets of $S$; for 2-element subsets one has $\dim(V_S) = 3$ while $\sum d_i = 4$; $S = [3]$ is similar.) But (i) fails since for $S = [3]$ one has $c_S = 2$ whereas $\sum_{i \in S} c_i = 3$.

