How do I find the kernel of a composition of functions? Functions $g$ and $f$ are linear and injective. How do I go about finding the kernel of $g \circ f$?
I'm asking because I want to prove that $\ker(f) = \ker(g \circ f)$.

One has $\def\Im{\operatorname{Im}}\ker(g\circ f)=\{\, v\mid g(f(x))=0\,\}=\{\, v\mid f(x)\in\Im(f)\cap\ker(g)\,\,\}=f^{-1}[\Im(f)\cap\ker g]$ whereas $\ker(f)=f^{-1}[\{0\}]$. Thus $\Im(f)\cap\ker(g)=\{0\}$ is a sufficient condition for being able to conclude that $\ker(g\circ f)=\ker(f)$ (which is of course not valid in general, as you question seems to suggest; just take $f=I$ to convince yourself of that). One can see that the mentioned condition is also necessary to get $\ker(g\circ f)=\ker(f)$: if one would have $0\neq w\in\Im(f)\cap\ker(g)$ and this means (because $w\in\Im(f)$) that there exists $v$ with $f(v)=w$; for this vector $v\in\ker(g\circ f)$ but $v\notin\ker(f)$.
