Finding some non zero endomoprhism $f$ satisfying $Kerf = Imf$ > My example: Find some non zero endomoprhism $f$ such that $Kerf = Imf$. * Maybe this question has been already answered somewhere on this page, but I have found it, then sorry. I just struggle to imagine this transformation. The vectors from the kernel always give me the zero vectors after a linear transformation, then how can I satisfy this equation $Kerf = Imf$, if $Imf$ is nonzero? I tried to use the matrix mapping: $f_A: V \to V$, $\;$ $f_A(x) = Ax$, $Ker(f_A) = \\{x \in \Bbb F^n, Ax = 0 \\}$ $Im(f_A) = \\{y \in \Bbb F^m, Ax = y \\}$ No idea what should I do next, please help me.

For example: $f: \mathbb R^2 \to \mathbb R^2$, $(a, b) \mapsto (b, 0)$. $\ker f = \mathrm{Im}\, f = \\{(a, 0) | a \in \mathbb R\\}$.

Or did I misunderstand the question?