I am analyzing the null space of a matrix $E$ given by
$E = \begin{bmatrix}AB \\ AC\end{bmatrix}$
where $E \in \mathrm{R}^{2m \times n}$, $A \in \mathrm{R}^{m \times n}$, $B\in \mathrm{R}^{n \times n}$, and $C\in \mathrm{R}^{n \times n}$.
Generally, we can say that the null space of B satisfies $\mathcal{N}(B) \subseteq \mathcal{N}(AB)$ and $\mathcal{N}(C) \subseteq \mathcal{N}(AC)$.
(1) If $A$ is a full row rank matrix,
can we say that $\mathrm{dim}\ \mathcal{N}(B) = \mathrm{dim} \mathcal{N}(AB)$ and $\mathrm{dim}\ \mathcal{N}(C) = \mathrm{dim} \mathcal{N}(AC)$ ?
(2) Can we say that the null space of E always satisfies $\mathcal{N}(E) = \Big( \mathcal{N}(AB) \cap \mathcal{N}(AC) \Big)$ ? Does equality make sense?
(3) If $\mathrm{dim}\ \mathcal{N}(B) \leq \mathrm{dim}\ \mathcal{N}(C)$ and $A$ is a full row rank matrix,
can we say $\mathrm{dim}\ \mathcal{N}(E) \leq \mathrm{dim}\ \mathcal{N}(AB)$ ?
Thanks for your time.