Let $V$ be a finite dimensional vector space, and $T_1,T_2$ be linear operators on $V$. When does it hold that $\text{ker}(T_1)\cap\text{ker}(T_2)\cong V/\text{span}\{\text{im}(T_1),\text{im}(T_2)\}$? Here ker and im denote kernel and image resp.
An application of this problem is as follows. Let $A$ be an f.d. semisimple associative algebra. Let $V$ be an f.d. right $A$-module, and $W$ an f.d. left $A$-module. Denote $V\otimes_A W:=V\otimes W/\text{span}\{va\otimes w-v\otimes aw:v\in V,w\in W,a\in A\}$. How can we prove that $V\otimes_A W\cong \{x\in V\otimes W:(a\otimes I-I\otimes a)\cdot x=0\}$? Here $I$ is identity operator.
Thanks!