Question 110
Let $m, n$ be integers, $1 \le m \le n$, $A \in \mathcal{M}_{m,n}(\mathbb{C})$, $B \in \mathcal{M}_{n,m}(\mathbb{C})$ and $f_{A,B} : \mathcal{M}_n(\mathbb{C}) \to \mathcal{M}_m(\mathbb{C})$, $f_{A,B}(X) = AXB$; $f_{B,A} : \mathcal{M}_m(\mathbb{C}) \to \mathcal{M}_n(\mathbb{C})$, $f_{B,A}(Y) = BYA$.
Prove that the following affirmations are equivalent:
1) The function $f_{A,B}$ is surjective.
2) The function $f_{B,A}$ is injective.
Prove that the following affirmations are equivalent:
1) The function $f_{A,B}$ is surjective.
2) The function $f_{B,A}$ is injective.