Theorem 1   A linear map between vector spaces is injective if and only if its kernel is $\{0\}$ .

Proof. Let $L: V \to W$ be a linear map. Suppose $L$ is injective, and $L(v)=0$ for some vector $v\in V$ . Also, $L(0) =0$ because $L$ is linear. Then $L(v)=L(0)$ , so $v=0$ . On the other hand, suppose $\operatorname{ker}L=\{0\}$ , and $L(v)=L(v')$ for vectors $v,v'\in V$ . Hence $L(v-v') = L(v)-L(v') = 0$ because $L$ is linear. Therefore, $v-v'$ is in $\operatorname{ker}L=\{0\}$ , which means that $v-v'$ must be $0$ .