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

Consider what it means to be injective and what it means to have a nontrivial kernel. Its not possible to have both. So, if the kernels are trivial then $\ker (f) =\{0\}$ and $\ker (g\circ f) =\{0\}$  However, you still need to check that that the $0$ in both of those is the same $0$ vector, ie that they are in the same vector space.
