Question

Consider the zero mapping O:U tends to V
defined by 0(v)=0 for all v belongs to V. Find the kernel and image of O

Answer 1

$O:U\to V \text{ such that } O(u)=0 \text{ for all } u\in U$

$Ker(O):=\{u\in U|O(u)=0\}$

By the definition of $O$, we see that $Ker(O)$ is the entire vector space $U$ since $O(u)=0 \text{ for all } u \in U$

$\implies Ker(O)=U$

$Img(O):=\{v \in V|O(u)=v\}$

By definition of $O$, we see that all vectors in the vector space $U$ maps to only a vector in the vector space $V$ which is $0_v$

Hence, $Img(O)=\{O_v\}$