Image equal to kernel Define the general formula of linear transformation $f:\mathbb{R}^{4}\rightarrow \mathbb{R}^{4}$ such that $Imf=Kerf$. Any ideas?--prophetes.ai

Image equal to kernel Define the general formula of linear transformation $f:\mathbb{R}^{4}\rightarrow \mathbb{R}^{4}$ such that $Imf=Kerf$. Any ideas?

**Hint** : Use the well known rank-nullity theorem $\text{dim(Ker(}f\text{))+dim(Im(}f\text{))=dim(}\mathbb{R}^4)$ to establish that (since the kernel is equal to the image) the dimension of both the kernel and the image is $2$. Can you pick it up from here?