Let's say you have to different representations of the same polyhedron $P\neq \emptyset$:
$$P=\{x\in \mathbb{R}^n\;|\;h_i^Tx\leq c_i, i=1,...,k \} =\{x\in \mathbb{R}^n\;|\;g_j^Tx\leq d_i, j=1,...,l \}. $$
I'd like to show that given those representations, if I know that the vectors $h_1, . . . ,h_k$ are spanning $\mathbb{R}^n$, then so do also the vectors $g_1, ..., g_l$.
How would you do that ?
I was thinking about using the properties of the polyhedron, seen as geometric objects, but I ended up showing nothing.