The kernal of a linear map, is the linear subspace of the dmain of the map which is mapped to zero vector.

Given $L:V\to W$, the kernal of $L$ is the vector space of all elements $v\in V$ such that $L(v)=0\in W$.

$$ \ker (L)=\{v\in V\mid L(v)=0\in W\}=L^{-1}(0\in W) $$

We get:

$$ \omega=L(v),K\in [\ker(L)=L^{-1}(0)], $$

image of a linear map

two elements of $V$ have the same image in $W$ iff their differnce lies inthe kernal of $L$, that is {note that linear map L}

$$ L(v_1)=L(v_2)\ \text{iff. }\ L(v_1-v_2)=0 $$