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 $\ker(f) = \ker(g \circ f)$.

If $g$ is injective then $\ker(f) = \ker(gf)$ is certainly true.  To prove two sets are equal you show that each is contained in the other.  For example if $x \in \ker(f)$ then $f(x) = 0$.  But then $gf(x) = g(0) = 0$ so $x \in \ker(gf)$.  This proves $\ker(f) \subseteq \ker(gf)$.  I'll leave the other direction to you (that direction uses the fact that $g$ is injective).
So if $g$ is injective $\ker(gf) = \ker(f)$ and if $f$ is injective then $\ker(f) = 0$ so if both are injective then $\ker(gf) = 0$.  This of course makes sense because the composition of two injective maps is again injective and hence has trivial kernel.
