Image equal to kernel

Question

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

Answer 1

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?