I believe that there are counterexamples for the following statements:
Let $V$ be a vector space and let $V_1$ and $V_2$ be subspaces.
- $V\cong V_1\times V_2 \Rightarrow V=V_1\oplus V_2$
- $\phi:V\rightarrow V$ linear operator $\Rightarrow V\cong \ker \phi\times I m \phi$
I am not so sure about the first statement. I know that $ V=V_1\oplus V_2$ if and only if the map $\psi:V_1\times V_2\rightarrow V_1\oplus V_2$ defined by $(v_1,v_2)\mapsto v_1+v_2$ is an isomorphism, but the statement for (1) above is not exactly this last statement I wrote.
I see that (2) holds if $\phi$ is invective, but I can't find a counterexample for the general statement. Finding counterexamples has never been my strong suit. Thand you for your help.