Counterex. for 1. $V\cong V_1\times V_2 \Rightarrow V=V_1\oplus V_2$ and 2. $\phi:V\rightarrow V$ lin op $\Rightarrow V\cong \ker \phi\times im\phi$

Question

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.

1. $V\cong V_1\times V_2 \Rightarrow V=V_1\oplus V_2$

and

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.

Answer 1

For your first statement, there is indeed a counterexample (with $V_1,V_2 \subseteq V$, moreover). Take $$ V_1 = V_2 = \{(t,0): t \in \Bbb R\} \subset \Bbb R^2 = V $$ We have $V \cong V_1 \times V_2$, but $V \neq V_1 \oplus V_2$. Look at the definition of $\oplus$ to verify that we indeed have $V \neq V_1 \oplus V_2$.

For your second statement: there is no counterexample. If you are only working on finite dimensional vector spaces, the rank-nullity theorem is sufficient here. Otherwise, recall that with the first isomorphism theorem, we have $$ V/ \ker \phi \cong \operatorname{Im} \phi $$ and $V = (V/\ker \phi) \oplus (\ker \phi)$