Consider distributions (generalised functions) with compact support $u_{ij} \in \mathcal{E}'(\mathbb{R}^n)$ for $1 \leq i, j \leq m$. Write $u_i = (u_{1i}, u_{2i}, \dotso, u_{mi})$ for the column vectors determined by $u_{ij}$ (vector valued distributions). Say $u_1, \dotso, u_m$ are weakly linearly dependent (WLD) if $$\det(u_{ij}(f)) = 0, \quad \mathrm{for \,\, all} \quad f \in C_0^\infty(\Omega).$$
Question: does WLD imply that $u_1, \dotso, u_m$ are linearly dependent? Hints toward references also greatly appreciated.
EDIT 2: the question as stated is clearly false: just take distributions with disjoint supports to see it's false. Thus it's natural to assume the supports of $u_{ij}$ are the same. However, then if we take n = 1 and $$u_{11} = \delta_0',\,\, u_{21} = \delta_0',\,\, u_{12} = \delta_0, \,\,u_{22} = \delta_0,$$ then $u_1$ and $u_2$ are WLD, but not LD (rows are LD, though). So the reasonable test case and a new question seems to be
Question: let $m = 2$, $u_1$ and $u_2$ have equal supports and are WLD. Then either $u_1$ is obtained by applying a constant coefficient diff op to $u_2$ or we obtain $u_2$ from $u_1$ in a similar way; OR we can say the same for rows.
This seems to be true for $n = 1$ and distributions supported at a point.