Past exam of the mathematical methods I course of the University of Oxford / 2017 / 9 / a / Solution

A map between vector spaces over the same field is linear when $L(av+bw)=aL(v)+bL(w)$ for every $v,w$ and scalars $a,b$. Its kernel is $\ker L=\{v\in V:L(v)=0\}$. It contains zero because $L(0)=0$. If $v,w\in \ker L$, thenThus it is closed under linear combinations and is a vector subspace of $V$. No injectivity or equality of the dimensions of $V,W$ is needed.