Could someone tell me what I am missing here?
I want to show that for linear maps $L_1, L_2$, $v\in \ker(L_1)+\ker(L_2) \implies (L_1+L_2)v = 0$.
$v\in \ker(L_1)+\ker(L_2)$
$\implies \exists a,b\in V : L_1(a)=0=L_2(b)\wedge v=a+b$
$\implies (L_1+L_2)v = L_1(a)+L_2(a)+L_1(b)+L_2(b)$
$\implies (L_1+L_2)v = L_2(a)+L_1(b)$...
But then...? Thanks.