Isomorphism theorem for several linear mappings

Question

Let $V,W$ be two (finite dimensional) vector spaces, $T_1,T_2$ be two linear operators from $V$ to $W$. Can we show that $V/(\ker T_1\cap\ker T_2)\cong \text{span}\{\text{img }T_1,\text{img }T_2\}$? Here ker and img denote kernel image of operator respectively. How about $T_1,\cdots,T_n$ instead of $T_1,T_2$ similarly? Thanks!

Answer 1

No. Let $V=W=\mathbb R^2$, $T_1(x,y)=(x,0)$ and $T_2(x,y)=(x+y,0)$

Then $\text{span}\{\text{img }T_1,\text{img }T_2\}=\{(t,0): t \in \mathbb R\}$ and $\ker T_1\cap\ker T_2)=\{0\}$.

Hence $\dim (V/(\ker T_1\cap\ker T_2))=2$, but $ \dim (\text{span}\{\text{img }T_1,\text{img }T_2\})=1$

Answer 2

To get some sort of result you could look at the map $$T_1 \oplus T_2 : V \longrightarrow W \oplus W, \; \; T_1 \oplus T_2 (v) = (T_1(v),T_2(v))$$ and apply the usual isomorphism theorem. Note that $\mathrm{ker}(T_1 \oplus T_2) = \mathrm{ker}(T_1) \cap \mathrm{ker}(T_2).$

So there is an isomorphism from $V / \mathrm{ker}(T_1) \cap \mathrm{ker}(T_2)$ to $\mathrm{im}(T_1 \oplus T_2).$

Unfortunately this is not clearly related to $\mathrm{span}(\mathrm{im}, T_1, \mathrm{im}\, T_2).$ @Fred gave a counterexample where the two have different dimensions.