Let $A \in M_{m \times n}(\mathbb{R})$ be of rank $m$. Choose the correct statement(s) from below: (A) The map $\mathbb{R}^n \longrightarrow \mathbb{R}^m$ given by $v \mapsto Av$ is injective; (B) There exist matrices $B \in M_m(\mathbb{R})$ and $C \in M_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$; (C) There exist matrices $B \in \mathrm{GL}_m(\mathbb{R})$ and $C \in \mathrm{GL}_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$; (D) For every $(B, C) \in M_m(\mathbb{R}) \times M_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$, $C$ is uniquely determined by $B$.
Let $A \in M_{m \times n}(\mathbb{R})$ be of rank $m$. Choose the correct statement(s) from below:\\
(A) The map $\mathbb{R}^n \longrightarrow \mathbb{R}^m$ given by $v \mapsto Av$ is injective;\\
(B) There exist matrices $B \in M_m(\mathbb{R})$ and $C \in M_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$;\\
(C) There exist matrices $B \in \mathrm{GL}_m(\mathbb{R})$ and $C \in \mathrm{GL}_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$;\\
(D) For every $(B, C) \in M_m(\mathbb{R}) \times M_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$, $C$ is uniquely determined by $B$.