Find all linear operators $ \widehat{A}\in\mathbb{V}_2$ which satisfy $\operatorname{ker}{\widehat{A}} = \operatorname{im}{\widehat{A}}$ Well, the point is to find all linear operators $\widehat{A}$ that satisfy the following conditions:


*

*$\operatorname{ker}{\widehat{A}} =  \operatorname{im}{\widehat{A}}$ and 

*$\widehat{A} : \mathbb{V}_2  \longrightarrow \mathbb{V}_2, \widehat{A}  \ne \widehat{ \mathcal{O} }$
where $\widehat{ \mathcal{O}}\left(\overline{x}\right) = \overline{0}, \, \, \, \forall \, \, \overline{x} \in \mathbb{V}_2$.


Hmm... As I see, for both $\operatorname{ker}\widehat{A}$ and $\operatorname{im}\widehat{A}$ $\operatorname{dim}=1$
But...I have quite no idea what to do.
Any help? 

Take a basis $\{u,v\}$ for $\mathbb{V}_2$, and define your operator as
$$\hat{A}u=v$$
$$\hat{A}v=0$$
Now you need to prove two things:


*

*Every choice of basis gives an operator that satisfies your conditions.

*Every operator that satisfies your conditions may be represented in this manner as a basis.

